CalGeo: Teaching Calculus using dynamic geometric tools
Outcome 1.4.1 ‘’Literature review about student understanding of topics in Calculus’’.
STUDENT UNDERSTANDING OF TOPICS IN CALCULUS
Introduction
For most students in mathematics, and science, calculus is the entry-point to undergraduate
mathematics. Because of its importance in such a wide range of disciplines, and its size of
enrolment, there have been many research studies in the student understanding of calculus.
The studies indicate that students enrolled in the traditional university calculus class have a
very superficial and incomplete understanding of many of the basic concepts in calculus.
There has been much concern in the failure to develop a conceptual understanding of calculus
topics because of the rote, manipulative learning that takes place in an introductory course
(Cipra, 1988; Steen, 1988; White & Mitchelmore, 1996). Romberg and Tufte argue that
students view mathematics as a static collection of concepts and skills to be mastered one by
one. Students are required to solve, sketch, find, graph, evaluate, determine, and calculate in a
straightforward fashion (Ferrini-Mundy & Graham, 1991). They are rarely engaged in "higher
level" problems. Lithner (2003, 2004) describes how most exercises in undergraduate calculus
textbooks may be solved by mathematically superficial strategies, often without actually
considering the core mathematics of the book section in question.This failure of the traditional
calculus curriculum has led to the calculus reform effort. Because of the importance of
calculus in many disciplines there has been a great deal of research into student learning of
calculus. The existing research provides a lot of implications for curriculum development and
reform.
This paper provides an outline of the current research in student understanding of topics in
calculus. The intent of the paper is to provide an overview of specific difficulties based on
education research. The topics presented are:

Functions and Variables

Limits and Continuity

Infinity
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
Derivatives

Integrals
References to various articles are provided for a more detailed discussion.
Functions and Variables
The function concept is a complex, multifaceted idea whose power and richness permeate
almost all areas of mathematics. In the school curriculum, function is an advanced topic that
is typically not explored in detail until the secondary level. Because of the function concept’s
unifying role in mathematics and its robust ability to provide meaningful representations of
complex real-world situations, particularly with the aid of graphing technologies (Heid,
Choate, Sheets, & Zbiek, 1995; Wilson & Krapfl, 1995), current reform recommendations
call for a functional emphasis to be integrated throughout the school curriculum, beginning in
the elementary grades (NCTM, 1989). Given the importance of functions in mathematics and
the curriculum, it is crucial for researchers to explore the nature of mathematics students’
knowledge of functions.
The idea of a function underlies many central concepts in calculus such as limit, derivative,
and integral. The importance of the idea of a function has led to much research in student
understanding of this topic. There are a number of papers, specifically those by Dreyfus,
Eisenberg, Vinner, and Monk, which deal exclusively with student understanding of the
function. The papers by Dreyfus and Vinner, and Monk also contain questions that may be
used to assess student performance.
Researchers have shown that many students have a primitive concept of a function. These
students enter a calculus course unable to provide a general definition of a function and when
prompted are only able to provide examples of functions. Even students who have a
Dirichelet-Bourbaki definition, describing a correspondence between two non-empty sets,
revert to a more primitive definition when asked certain types of questions.
Dreyfus and Eisenberg found in interviews that some students state that a relation is a
function only when it can be represented by a single formula (Ferrini-Mundy & Graham,
1991). Vinner and Dreyfus show that even when students have a Dirichelet-Bourbaki
definition of a function, their behaviour might be based on the formula conception when
working on identification or construction tasks. Their study was performed on several groups
of first-year college students in two Israeli institutions and some junior high school
mathematics teachers who had not majored in mathematics. Their work showed that many
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who gave the Dirichelet-Bourbaki definition for the function concept later used a formula
definition stating that a particular graph could not be a function because "two parts of [the]
graph . . . have different characters [and therefore] cannot be described by a single formula
(Vinner & Dreyfus, 1987)."
The study done by Vinner and Dreyfus (1987) as well as the study by Christou et al. (2005)
also showed some students rejected certain graphs of functions because of a correspondence
being discontinuous at one point or a correspondence having a point of exception. The study
also probes student understanding in construction tasks and shows that "students usually pay
less attention to the conceptual aspects of a given notion and more attention to its
computational or operational aspects (Vinner & Dreyfus, 1987).
The interviews by Dreyfus and Eisenberg (1982) also show that students view algebraic data
and graphical data as separate; a graphical representation with no formula has no meaning for
most entering calculus students. Heid (1988) points out that traditional course development of
graphs requires only that students master the methods of using zeros of derivatives to sketch
graphs of quadratic and cubic functions and that they use the general notion of checking
functional behavior near undefined function values for a small, well-defined set of functions
whose graphs have linear asymptotes. Typically, instructors confine graphing assignments to
requiring a sketch for a graph like that of
or
or
They seldom ask students to draw conclusions on the basis of their graphs
or to comment on the relationship between two different graphs (Heid, 1988).
In her experimental class taught to college students, Heid stresses instruction which
encourages students to reason deeply from and about graphs using computer technology.
Most traditional calculus courses offer little opportunity for students to develop a deep
conceptual understanding of the graph and do not promote an understanding of the connection
between an algebraic representation and a graphical representation.
Students also feel that when they are given a function they are expected to do something with
it, such as substitute a value (Graham & Ferrini-Mundy, 1989). They view the function as a
static quantity, thinking of only one point at a time according to Monk. Monk's study shows a
distinction between a "point-wise understanding" and an "across-time understanding." A
point-wise view considers the function as a correspondence between two sets or between two
variables whereas an across-time view deals with how a change in one variable leads to a
change in others. An across-time understanding requires the student to view the function at
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infinitely many points (or with a continuously changing variable), instead of one point at a
time. The study indicates "that, at least in simple situations, students have a confident and
secure Point-wise understanding of functions, but even at the end of the first or second quarter
of calculus . . . they are still struggling to use functions in an Across-Time manner (Monk,
1987)." This may be partially due to the point-wise view being presented in calculus and precalculus texts in their introductions of the concept of a function, even though an across-time
view is more applicable to topics in calculus. Monk's study also indicates that the prerequisite for an across-time understanding is a point-wise understanding, although "it does not
seem to be the case that an Across-Time understanding comes easily and automatically after a
Point-wise understanding has been developed (Monk, 1987)." This seems to be the case in
three of the four questions analyzed by Monk.
White and Mitchelmore (1996) showed that students have a primitive understanding in the
concept of a variable. The study involved four questions and each question had four versions.
Version A required the students to do more translation from English to mathematical
symbolism while version D required them to do very little translation. These questions were
used in their research performed on first year university students, all of whom had studied
calculus in secondary school. They found that students treated variables as symbols to be
manipulated rather than as quantities to be related. In the problems that were given, the
students were unable to distinguish between a general relation and a statement of a specific
variable. This underdeveloped concept of a variable made it difficult to identify and
symbolize an appropriate variable by translating one or more quantities and therefore define a
usable function.
Others (Judson, & Nishimori, 2005) have investigated the difference in concepts and skills in
High School Calculus between Japanese and American students. There results showed that
there very little difference in the conceptual understanding of the two groups of students but
the Japanese students demonstrated much stronger algebra skills than their American
counterparts.
Berry and Nyman (2003) also confirmed students’ algebraic symbolic view of calculus and
the fact that they find it difficult to make connections between the graphs of a derived
function and the function itself.
Rozier and Viennot (1991) also see students treating variables in a primitive manner. Their
study showed how students reduce the number of variables, or take all the variables into
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account, but in a simplified way, when dealing with thermodynamic problems. One way the
students reduce the number of variables is in their "tendency in coping with multivariable
problems . . . to forget some relevant variables (Rozier & Viennot, 1991). Because of a
"preferential association" between two variables, we see students relating only these two
variables and ignoring the others. Rozier and Viennot (1991) also see students reducing the
number of variables by combining two variables and treating them as one. Linear causal
reasoning is another way students are able to deal with only two variables at a time even when
the "changing physical quantities are all supposed to change simultaneously (Rozier &
Viennot, 1991)." This linear reasoning involving successive steps allows the student to relate
two variables while keeping the others constant during each step. Thinking about more than
two variables at a time seems to be a very difficult task for students and this difficulty
surfaces in various topics such as thermodynamics.
In their work Carlson, Jacobs, Coe, Larsen, and Hsu (2002) also discuss students’ difficulty in
moving from a coordinated image of two variables changing in tandem to a coordinated
image of the instantaneous rate of change with continuous changes in the independent
variable for dynamic function situations.
Berry and Nyman (2003) also confirmed students’ algebraic symbolic view of calculus and
the fact that they find it difficult to make connections between the graphs of a derived
function and the function itself.
Sajka (2003) presented a case study on a secondary school students understanding of the
concept of function. Her analysis was based on the procept theory (Gray & Tall, 1994). The
author identified three main sources of the difficulties. The intrinsic ambiguities in the
mathematical notation, the restricted context in which some symbols occur in teaching and the
limited choice of mathematical tasks at school and thirdly the idiosyncratic interpretations
that the individual was giving to the tasks.
Definition and image of the function concept. A useful way to characterize a person’s thinking
about functions is in terms of the notions of concept definition
and concept image. These constructs point to a distinction between the formal definition an
individual holds for a given concept and the way that he or she thinks about the concept.
Usually when a concept name is seen or heard, what is evoked is not the formal definition but
the concept image consisting of “the visual representations, mental pictures, the impressions,
and the experiences associated with the concept name” (Vinner, 1991, p. 68). A person can
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hold a concept image for function that does not seem to correspond directly to a formal
mathematical definition, as found in the investigation of students’ learning of functions by
Vinner and Dreyfus (1989).
Because of developments within mathematics, over the years the generally accepted notion of
function has changed from a means to describe dependence relationships between quantities
to the highly abstract mathematical construct that now comprises the modern conception of
function (Cooney & Wilson, 1993; Dreyfus, 1990). Since the 1930s the official definition of
function (known as the Dirichlet-Bourbaki definition) has emphasized univalent
correspondence over covariation, largely because of the many problems this definition can be
employed to solve and the numerous useful relationships it can be used to describe,
relationships not possible to describe under the more restrictive covariation definition
(Cooney & Wilson, 1993; P. W. Thompson, 1994; Vinner & Dreyfus, 1989). Analogous
development has occurred in the treatment of the function concept in school textbooks in this
century (Cooney & Wilson, 1993), with the Dirichlet-Bourbaki definition currently
dominating most high school curricula (Dreyfus, 1990).
The evolution of the function concept is often portrayed as a move from an operational notion
as a process to a structural notion as an object (Sfard, 1991). A body of recent research, much
of it applying the process-object distinction, supports the idea that the historical evolution of
the function concept may provide a pedagogical sequence that better supports the students’
development of deep conceptual understandings of functions (Eisenberg, 1991; Markovits,
Eylon, & Bruckheimer, 1986; Sfard, 1991; Thompson, 1994; Vinner & Dreyfus, 1989).
Recent calls for reform have begun to reflect a growing consensus that a covariation approach
furnishes an indispensable first step in the direction of a deep comprehension of the function
concept in general (NCTM, 1989). In light of the historical tension between covariation and
correspondence in the school curriculum and in students’ understandings of functions, it is
advantageous for teachers to recognize and appreciate features of both definitions.
Repertoire of functions in the high school curriculum. An important component
of a teacher’s conceptions is the ability to differentiate among various types of functions by
identifying their unifying or distinguishing characteristics (Even, 1990; Norman, 1992). A
teacher’s basic repertoire of functions should represent competence with the major families
that are encountered in the school curriculum: linear, polynomial, exponential and
logarithmic, and trigonometric. This requisite understanding encompasses an appreciation for
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the diversity of problem situations that can be modeled by a given type of function.
Additionally, teachers should have access to a wide range of examples that serve to illuminate
or counter specific properties of functions. A teacher’s repertoire of examples and problem
situations for functions is one of the key bases from which meaningful instruction may be
constructed.
Some researchers have investigated the way in which functions appear in Middle School
Textbooks. For instance Mesa (2004) analysed the exercises and problems about functions in
middle-school textbooks from 15 countries using Balacheff’s theory of conceptions and
Biehler’s notion of the prototypical domain of application of concepts in order to describe the
practices associated with the notion of function. The analysis yielded five different practices –
symbolic rule, ordered pair, social data, physical phenomena and controlling image.
The importance and use of functions in varying contexts. Valuable conceptions of function for
teaching include the capacity to use and appreciate the importance of the concept in areas
other than mathematics and in different contexts within mathematics itself (Markovits et al.,
1986; Norman, 1992). Teachers need to have this type of understanding not only to answer
students’ questions (e.g., “Why are we doing this?”) but also to promote, through instructional
actions, the power and possibilities of the function concept. A crucial feature of
comprehensive conceptions of functions is a recognition of changes in the surrounding world.
Equally important is an identification of the relationships among those changes as tools for
making sense of them (Dreyfus, 1991; Sierpinska, 1992). Recent reform recommendations
(NCTM, 1989) and reform curriculum materials (Core-Plus, 1994) have emphasized
modeling and contextualized explorations of mathematical relationships.
Multiple representations and connections among them. Because of the complex nature and
manifold uses of the function concept, functional situations lend and verbal descriptions. As
new technology facilitates more efficient constructions, multiple linked representations are
increasingly finding their way into the school curriculum. An understanding of functions in
one representation will not necessarily correspond to an understanding in another
representation, but ability to translate among varied formats is necessary to effectively
interpret problem situations. When combined, the information gleaned from diverse
representations contributes to a deeper, more comprehensive understanding of the under-lying
functional situation (Even, 1990). According to Kaput’s (1989) theory of linked
representation systems, the common representations of functions form the basis of a concept
image. Because an individual can develop multiple concept images, which can exist in both
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complementary and contradictory ways, the more tightly connected the representations, the
more robust and compatible the system of concept images will be. An integrated concept
image that highlights the associations among representations and facilitates flexible moves
among them is particularly beneficial in that “one has control over the representations one
wants to use” (Dreyfus, 1991, p. 39). Because each representational format has varying
limitations or strengths in different contexts, it is beneficial to have the choice of which
representations to employ and the knowledge needed to make such a choice.
A number of researchers have investigated students’ abilities in regard to visual aspects of the
calculus concepts:
Bremigan (2005) has investigated the diagrams that mathematically capable high school
students produced in solving applied calculus problems in which a diagram was provided in
the problem statement. The results provided insight into the various ways that students
modified these diagrams and constructed new ones. The research has also investigated the
relationship between the frequency or nature of the diagrams produced by high- and lowscoring students and also their problem solving success. Her results show that females who
were less successful in problem solving, produced more diagrams than males. Diagrams
constructed or modified by males tended to be simpler than the more elaborate versions
produced by females.
Borgen and Manu (2002) have investigated the understanding of students who on paper have
presented a picture-perfect solution to a calculus problem. Their results showed that these
picture perfect solutions are may be accompanied by both understanding and
misunderstanding.
A number of pieces of research have concentrated on students’ intuitions in regard to calculus
concepts. Recently Zazkis, Liljdedahl, and Gadowsky (2003) have investigated secondary
school students’ and secondary school teachers’ difficulties in function translations. The
participants focused on patterns, locating the zero of the function problems and the point-wise
calculation of function values. The results showed that the horizontal shift of the parabola was
inconsistent with the participants’ expectations and counterintuitive for most of them.
A not so common topic of investigation in regard to linear functions was Karsenty’s
investigation into what adults remember from their high school mathematics. In general the
findings of this paper support the idea that retaining high school mathematical content
strongly depends on the number, level and total length of mathematics courses taken by the
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students. The researchers analysed three cases based on recall theories in order to explain the
mechanism of recalling in terms of reconstruction vs reproduction.
THE APOS THEORY
The Action Process Object Schema (APOS) theory has been illustrated and elaborated in a
number of papers (see, e.g., Asiala et al., 1996; Asiala, Cottrill, Dubinsky, & Schwingendorf,
1997; Clark et al., 1997). Schemas, in particular, served a significant role in this study. A
schema is a coherent collection of actions, processes, objects, and other previously
constructed schemas that are coordinated and synthesized by the individual to form structures
utilized in problem situations. An individual demonstrates the schema’s coherence by
discerning what is contained within the schema and what is not. The individual may reflect on
a schema and transform it as an object to perform new actions. Through this transformation,
the schema itself can become an object. Objects can be transformed by higher level actions,
leading to new processes, objects, and schemas to construct new concepts. Hence, the action,
process, and object development continue to be reconstructed in existing schemas.
Schema Development
Schema development is a dynamic, ever-changing process. Piaget and Garcia
(1983/1989, 1996) proposed that knowledge grows according to certain mechanisms and that
it evolves in three stages, called the triad, that occur in a fixed order.
The nature of the triad stages is functional, not structural, and we describe the triad’s general
psycho-dynamical aspects. Piaget and Garcia hypothesized that these levels can be found
when one analyzes any developing schema.
In the preliminary phase of the triad, the intra level, particular events or objects are analyzed
in terms of their properties. Explanations at this level are local and particular. An object in the
intra level is not recognized by the learner as necessary, and its form is similar to the form of
a simple generalization. For example, at the intra level of the derivative schema, the student
interprets the function as the correspondence of points in a graph, demonstrating this
interpretation with the recognition of stereotypical, simple functions. At the intra stage the
student concentrates on a repeatable action or operation but lacks the capacity to relate the
action to a system of conditions through which he or she could extend its application and
include it in a system of interdependent transformations and representations.
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The student’s use of, comparison of, and reflection upon isolated ideas leads her or him to the
construction of relations, transformations and mainly representations in the inter level. In the
inter stage, the student is aware of the relationships present and can deduce from an initial
representation, once it is understood, other representations that are implied by it or that can
coordinate it with similar representations. This process leads the student to group the systems,
using a method that includes a new representation. For example, at the inter stage of the
function schema, the student coordinates the notion of symbolic representations with the idea
of the graphical forms of function representations.
When a student reflects upon these coordinations and relations, new structures evolve.
Through syntheses of the inter-level representations, the student constructs an awareness of
the completeness in the schema, and in the trans level, the student can perceive new global
properties that were inaccessible at the other levels. At the trans stage of the function schema,
for example, the student recognizes that all situations in which representation is involved are
related to the function concept. The student can discriminate between those relationships that
are included and those that are not, thus demonstrating coherence of the schema.
At each stage of the triad, the student reorganizes knowledge acquired during the preceding
stage. The progression is gradual and not necessarily linear. In the process of learning, one
develops different schemas, and the growth and change of each schema can be described
using the triad. As knowledge develops, people construct many coexisting schemas, all of
which are constantly changing and at varying levels of evolution. Therefore, in some problem
situations a person may need to coordinate different schemas. Each schema is itself made up
of actions, processes, objects, and other schemas and their relationships. In particular, one
schema may rely heavily on the development of one or more other schemas. In such cases,
one may be able to identify those component schemas and their multidimensionality.
Therefore, in understanding the development of the overall schema, one can identify not only
the component schemas’ growth but also their coordination. Within the overall schema this
coordination leads to new structures that are built on the properties of the component
schemas.
Using the triad as a tool with which to analyze the knowledge composition of students,
researchers are able to take into account the richness of problem situations by focusing
attention on relationships among ideas. Because a student may proceed through the levels in a
unique and nonlinear fashion, the triad, as a tool, allows researchers to use a detailed approach
in a flexible manner. It utilizes complexities of the problem, relationships among ideas, newly
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formed structures, and coherence—all important aspects of the development of logicomathematical structures.
Schema Development and APOS Theory
Building on the work of Piaget and constructivist ideas, Asiala et al. (1996) introduced the
APOS theory, in which they described an individual’s schema for a mathematical topic as the
totality of his or her knowledge connected (consciously or unconsciously) to that topic.
Furthermore, they stated that a schema can be included within a higher level schema of
mathematical structures. One may not necessarily access each of these constructions in all
situations, because mathematical learning is highly nonlinear. This theory of schema
development may explain why students have difficulty with different parts of a topic and may
even have different problems with the same situation on different encounters. These partial
understandings are attained and later added bit by bit with a continuing reorganization of
ideas.
The theory of a triad of stages in schema development, an integral component of APOS
theory, has been used in several studies of student understanding in various areas of
mathematics. In their study of student understanding of the chain rule, Clark et al. (1997)
found that the APOS theory involving actions, processes, and objects was not adequate for
analyzing their data on student understanding but that the triad of Piaget and Garcia
(1983/1989) was useful in interpreting the levels of understanding. The authors described
students at the intra stage as having a collection of rules for differentiation, including some
special cases of the chain rule, but not recognizing any relationships among these rules. At the
inter stage, the student begins to collect these special cases and to realize that they are related,
but the student is unaware of the general relationship. Finally, at the trans stage, a student has
constructed the underlying structure of the chain rule and can determine which instances are
and which are not part of this chain-rule schema.
In two recent papers Dubinsky, Weller, Mcdonalr and Brown (2005) apply the APOS theory
in order to suggest a new explanation of how people might think about the concept of infinity.
They propose cognitive explanations and in some cases resolutions, of various dichotomies,
paradoxes and mathematical problems involving the notion of an infinite process and certain
mathematical issues related to the concept of infinity. These explanations are expressed in
terms of the mental mechanisms of interiorisation and encapsulation.
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Several studies have documented that even students who are able to perform well on
routine calculus problems have difficulties on nonroutine problems (e.g., Selden, Selden, &
Mason, 1994). Some believe that these difficulties are due to a weak conceptual view of
function (Orton, 1983; Selden et al., 1994). Additionally, the ordinary learning and
reorganization of knowledge sometimes incorporates mathematically incorrect constructions
that are held for a period of time. Generally the ability to visualize in mathematics is thought
to be beneficial, but Aspinwall, Shaw, and Presmeg (1997) reported on a student whose
ability to think about a problem was hampered by an incorrect image. The student had
constructed an image of a second-degree polynomial function as having vertical asymptotes.
However, he asserted that the domain was all real numbers. This erroneous image caused him
to draw a graph of the derivative shaped like a cubic function. This drawing was in conflict
with the student’s analytic knowledge that the derivative of a quadratic function should be a
line. Nevertheless, he was unable to control his
Understanding functional properties as they relate to calculus learning has also been
studied. In his paper describing the growth-oriented view of function as object, Slavit (1995)
commented that students do not encounter some functional properties, such as cusps, until
they reach calculus courses. For these students, interiorization of function properties is limited
by the classes of functions they have studied. Slavit called for further investigation of the role
of functions in calculus courses. He also reported that the nearly exclusive study of
polynomial and exponential functions, which are smooth and continuous, led to
overgeneralizations about continuity.
Limits and Continuity
Confrey argues that students enter a calculus course in one of three ways: with a discrete
understanding of continuous ideas; with an independent transition to the idea of continuity; or
with an algorithmic approach (Ferrini-Mundy & Graham, 1989). Tall states that part of the
difficulty in the conceptual understanding of a limit is in the colloquial meaning of the terms
used when referring to limits. To many students the statement "we can make sn as close to s as
we want" means we can make sn close but not coincident. Other research, done by FerriniMundy and Graham in 1989, shows that when students are asked to evaluate limits of the
form lim f(x) as x a they are quite successful, but when asked for a geometric interpretation,
students showed very little understanding. This shows the independence of the algebraic and
graphical representations. One interview revealed that the notion of "approaching" was not
part of the student's understanding of the limit. The student stated that limit problems were
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simply functions to be evaluated and that "the graph can't help me find an answer (FerriniMundy & Graham, 1991)." Heid (1988) shows similar student difficulty in the understanding
of limits in two student populations at Penn State. One population was taught using graphical
and symbol manipulation computer programs while the other population was taught by more
traditional methods. Both populations seemed to "identify the notion of a limit with a process
rather than a number, focusing on the "getting close to" rather than on the number being
approached. This confusion about a limit permeated their explanations of derivatives as they
"described derivatives as approximations for slopes of tangent lines rather than as being equal
to the slope (Heid, 1988)."
Students often continue to insist that a function cannot reach its limit, even after they have
worked with many examples of functions that do reach their limits. They often view such
disconfirming examples as exceptions to valid general principles, and thus continue to hold
their original ideas even after being presented with the counter-examples.
Since the formal definition is in disagreement with students' dynamic, intuitive ideas of limit,
this persistence of their informal ideas means that students often are unable to make sense of
the formal definition, and may be unwilling to try. They may view the formal definition as a
useless bit of arcane trivia that their teachers insist on covering, rather than seeing it as an
expression of the essence of the idea of limit. In this case, students may respond based on
their informal dynamic ideas when asked about a particular example, and give a completely
different response when asked for the definition of limit. Such a student makes no effort to
reconcile the two ideas, and so often does not notice when the two responses are in direct
conflict with one another.
Teachers and textbooks may unintentionally reinforce students' preference for intuitive ideas
and disregard of formal definitions. Students tend to prefer strategies that they see as
practical, enabling them to complete their homework and attain acceptable scores on exams.
When homework and exams consist of repetitive problems that students can solve without
recourse to a formal concept or definition of limit, students are encouraged to regard formal
definitions and concepts as irrelevant. Since many homework problems can be solved by the
application of memorized procedures, even the informal concepts may be devalued in the
minds of students intent on completing the required problems with a minimum of effort.
Disagreement between formal definitions and informal concepts is only one example of a
situation in which a student may hold two mutually contradictory ideas and not notice a
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conflict. Even when the formal definition is not involved, students working with their
informal ideas may respond to essentially the same problem in different and contradictory
ways when it is presented in different forms. For example, although the sum of the infinite
series 0.9 + 0.09 + 0.009 + 0.0009 + ... , is the same as the limit of the sequence of partial
sums of the series, 0.9, 0.99, 0.999, 0.9999, ... , and also the same as the repeating decimal
0.9999..., students may say that one of these representations is equal to 1 while another is not.
Apparently, the students hold several concept images, and select one according to the form of
the question.
The wording of the question plays a role in the selection of the particular concept image that
the student brings into play when presented with a problem. Words and phrases that
mathematicians consider to be synonymous may have very different connotations for
students. To design instruction that will help students develop the desired concept image, one
must first understand their intuitive ideas and the language in which these ideas are expressed.
It is tempting to think of the students' informal ideas as stumbling blocks along the path
toward formal concepts and definitions, and indeed there is some truth in this. However, these
stumbling blocks are also the building blocks from which the more sophisticated concepts
will be constructed. No complicated concept is ever acquired in an instant. The formal
concept of limit must be built up through a process of forming, accepting, rejecting,
modifying, and connecting more primitive and intuitive concepts.
The paper by David Tall and Shlomo Vinner (1981) is primarily conceptual. In support of
their ideas, they include reports of some empirical studies not reported elsewhere. Since this is
intended as a review of empirical studies, I will examine the empirical work discussed in their
paper. Other than one brief mention of an interview with a group of four students, the work
reported here is primarily quantitative. Surveys were given to large groups of undergraduates,
36 in one study and 70 in another. Short answers to free-response questions were categorized,
and the numbers of each type were reported and discussed.
These studies all revolve around concept images (the student's image of the concept), concept
definitions (words used, either by the student or the teacher, to specify the concept), and
concept definition images (the student's image of the meaning of the concept definition). Tall
and Vinner write:
Many concepts we meet in mathematics have been encountered in some form or other
before they are formally defined and a complex cognitive structure exists in the mind
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of every individual, yielding a variety of personal mental images when a concept is
evoked. In this paper we formulate a number of general ideas intended to be helpful in
analysing these phenomena and apply them to the specific concepts of continuity and
limits.
In a questionnaire for mathematics students arriving at an English university, Tall asked
students to find the limit lim n->° (1 + 9/10 + 9/100 + 9/1000 + ... + 9/10n), give a definition
of limit, and say whether 0.999... (repeating) = 1 or not, giving a reason for their answer.
Although it seems clear that the requested limit must be 1.999..., which could be 2 only if
0.999... = 1, fourteen of the 36 students (39%) correctly gave the limit as 2, but incorrectly
said that 0.999... is less than 1. Tall and Vinner remark "Clearly the two questions evoked
different parts of the concept image of the limiting process."
In a subsequent test, the same students were asked to write several decimals as fractions,
including the repeating decimals 0.333... and 0.999... . Thirteen of the fourteen students who
had previously said 0.999... < 1 now said 0.999... = 1, often indicating in their responses an
awareness of cognitive conflict; they crossed out answers and wrote things such as "0.999... =
3 x 0.333... = 3 x 1/3 = rubbish." (page 159) Tall and Vinner use this to illustrate their point
about the significance of the difference between potential cognitive conflict, in which a
student holds two concept images that are inherently contradictory, and actual cognitive
conflict, in which the student calls on both concept images at the same time and becomes
aware of the contradiction. These fourteen students displayed potential cognitive conflict in
their results to the first questionnaire, but did not experience actual cognitive conflict until
presented with the questions on the second test.
While actual cognitive conflict may lead to refinement of the concept image and/or the
concept definition image, this is not always the case. Students who experience actual
cognitive conflict when presented with an anomalous example may preserve their sense of
equilibrium by discounting the example. Four students who had indicated that they believed
that no term of a series can be equal to the limit of the series were shown the example 0, 1/4,
0, 1/8, 0, 1/12, ..., which can also be written sn = {0 for odd n, 1/2n for even n}. In this series,
many of the terms are equal to the limit of 0. The students all concluded that this was not a
real series, but was two. The series 1/4, 1/8, 1/12, ... tended to zero, while the terms 0, 0, 0, ...
were equal to zero. Although the given series fits the formal definition of a series, it failed to
fit the students' concept image, so they discarded it, insisting that it was not really a series.
Tall and Vinner write: "This is a typical phenomenon occurring with a strong concept image
15
and a weak concept definition image that permeates the whole university study of analysis,
especially when there are potential conflict factors between the two." (page 160)
Finally, Tall and Vinner gave a questionnaire to seventy first-year university students, all of
whom had previously earned grades of A or B in A-level mathematics in English high
schools. First, the students were asked to explain what is meant by lim x->1 (x3 - 1) / (x - 1) =
3. On the next page, they were asked to write down a definition of lim x->a f(x) = c if they
knew one. Responses were categorized according to correctness and type, either formal or
dynamic. The 31 students who gave a dynamic definition all used a dynamic approach to the
example, as did most of the 21 students who did not give a definition. Of the four students
who gave a correct formal definition, three used a formal approach to the example, and one
used a dynamic approach.
All of this is as might be expected--students generally used the same approach to the example
as to the definition. However, there was one exception. Only four of the fourteen students
who gave incorrect formal definitions used their formal definitions in their approach to the
example. The other ten used a dynamic approach to explain the example. For these ten
students, the request for a definition apparently evokes a different concept image than is
evoked by a specific example. Since they gave incorrect definitions, it appears that their
concept definition image may have been weak or faulty. Their failure to use their concept
definitions on the example suggests that their concept definition images may be in conflict
with, or disconnected from, their concept images of limit.
Tall and Vinner go on to describe similar work with students' concept images and concept
definition images related to continuity. As with limits, the students may have a concept image
of continuity that works well for simple problems, but that is at best weakly connected to a
formal definition. Many have little or no concept definition image. The students often cling to
these informal ideas, creating an obstacle to their learning of the more formal concept and
preventing them from understanding how continuity relates to the unusual problems and
anomalous examples Tall and Vinner used in their study.
Tall and Vinner conclude by writing "The difficulty of forming an appropriate concept image,
and the coercive effects of an inappropriate one having potential conflicts, can seriously
hinder the development of the formal theory in the mind of the individual student."
Later on Tall and Pinto (1999) made a distinction between natural and formal thinkers.
Natural learners use their pre-existing intuitive understanding of mathematical concepts to
16
give meaning to the concept’s definition and associated formal work. Formal learners build
their concept by examining the logical entailment of a concept’s definition. More recently
Webber (2003) described a third type of students. He described procedural learners as learners
who “fist master flexibly procedures to write a class of proofs and only later reflect on these
procedures to give meaning to the techniques and concepts that they are studying. When
procedural learners first learn to write a class of proofs, they do not understand why these
arguments are valid proofs, but simply produce these arguments to receive credits on
assignments.” (Weber, 2004, p.129). Research has shown that all three types of students may
be successful or unsuccessful in their attempt to deal with calculus concepts.
John Monaghan (1991) studied the effects of the language used in teaching and learning
limits. Words taken from natural language, having familiar everyday meanings, are given
subtly different meanings in mathematics. Students may confuse the two. Monaghan writes:
The phenomena reported on here were part of a wider study which investigated
mathematically able adolescents' conceptions of the basic notions behind the calculus.
This paper reports only on those aspects of the study that examined students'
understanding of the language used by teachers to communicate calculus concepts. . . .
This article deals solely with ambiguities inherent in the four phrases tends to,
approaches, converges, and limit.
Monaghan administered a questionnaire to 54 English high school students who had passed
O-level exams in mathematics at the age of 16 and were in their first year of A-level studies.
Twenty-seven were studying A-level mathematics and 27 were not. A month after the
questionnaire, the subjects were interviewed. The following year, a revised form of the
questionnaire was administered to 190 students, 114 studying A-level mathematics, from
schools following the same O-level and A-level schedule as in the first sample. The subjects
in the second sample were not interviewed.
Several items with Likert-type responses (yes / think so / unsure / think not / no) were
common to both questionnaires. Each item asked students the same question in four different
ways, using the four phrases tends to, approaches, converges, and limit. One item asked about
the sequence of numbers 0.9, 0.99, 0.999, 0.9999, . . . with suggested limits of 1 and 0.999
repeating. Six other items related to continuous curves given by sketches. All questions were
asked about convergence to 0, although two of the curves actually converged to 1. The other
four did converge to zero. In addition to these items, students in the first sample were asked to
17
write four sentences, each illustrating the use of one of the four phrases. Students in the
second sample were asked to write sentences using the word limit, but not the other three
phrases.
Although a mathematician might say that the sentences: "The curve tends to 0," "The curve
approaches 0," "The curve converges to 0," and "The curve has 0 as a limit" all mean the
same thing, the students often agreed with one but disagreed with another in reference to the
same curve or sequence. For example, 66% of the second sample thought the given sequence
of numbers tended to 1, but only 22% thought it converged to 1.
Quantitative analysis of the Likert-type items indicated that the students understood these four
phrases to have very different meanings. Monaghan examined the responses to the interviews
and the sentence-writing task to determine the nature of these different concepts. Limit was
often thought of as a bound. Some examples involved legal bounds, such as speed limits. It is
possible to exceed the speed limit, although it is forbidden. Other conceptions of limit
involved a bound on physical or mental abilities, such as a limit to the height one could jump
or a limit to one's patience.
Approaches was seen in a more dynamic way. It involved things moving toward other things,
sometimes with the idea that the object being approached would eventually be reached ("The
train approached the station.") and sometimes with the suggestion that the moving object
might neither reach the object being approached nor even get particularly close. In the Likerttype items, the graph of the function f(x) = 1 + 1/x was viewed by about a third of the first
sample and half of the second as approaching 0, since the curve decreases as x increases, even
though it always remains above its asymptote at y = 1.
In responses to the sentence-writing task, nearly all examples of tends to involved personal
inclination, as in "he tends to wear jeans" or "she tends to drink a lot." In the Likert-type items
providing mathematical examples, students generally treated tends to as having the same
meaning as approaches. Both are seen as dynamic, evoking movement, and generally
referring to motion of an object that does not reach the point approached.
Converge was also seen as dynamic, but usually involved two continuous objects, as in "the
roads converge" or "the light rays converge." Other references involved discrete objects
making contact, such as "the footballers converged on the ball." One student said that two
sequences could converge to one another, as light rays might converge, but that it didn't make
sense to talk about one sequence converging. Also, converging more often involved reaching
18
or touching, rather than getting closer without touching, as was implied by tends to and
approaches.
Monaghan quotes Tall and Vinner (1981), who wrote: "We shall call the portion of the
concept image which is activated as a particular time the evoked concept image. At differing
times, seemingly different conflicting images may be evoked." (page 24 in Monaghan, page
152 in Tall and Vinner) Monaghan suggests that mathematically synonymous terms bring
different connotations from their natural language meanings, and thus play a role in evoking
different concept images for the same mathematical concept. In conclusion, Monaghan
recommends: "Students should be led to explore and discuss their own conceptions and to
realise how everyday meanings of mathematical phrases can direct them into fallacious
interpretations."
Steven Williams (1991) examined students' conceptions of limit and their responses to the
challenge to those conceptions posed by the presentation of anomalous examples. He writes:
In an effort (a) to explore in more detail what Cornu called students' spontaneous
models of limit, and (b) to study the means by which these models can be altered and
made more rigorous, the study investigated the views of 10 second-semester calculus
students as to what constituted a limit and then presented them with descriptions and
examples of limits that conflicted with their views.
This study was conducted under the assumption that most calculus students form
rudimentary limit models . . . and has sought specifically to (a) document and describe
the existence of such models, (b) attempt to use exposing and discrepant events to
encourage a change in students' understandings of limit so that they more nearly
resemble the formal definition, and (c) document the factors affecting such change.
The subset of data reported here focuses on two models of limit found to be most
common among students in the study.
Williams gave a brief questionnaire on limits to 341 students enrolled in second semester
calculus as a university. Students were asked to mark each of six statements about limits as
true or false, select which one of the statements they thought best described a limit, and write
a few sentences explaining what they thought a limit was. Williams presents a summary of the
results of the questionnaire, giving the proportion of students who selected each of the
suggested views of limit. From the results of this questionnaire, he selected ten subjects
whose questionnaires each clearly and unambiguously presented one of the four common
19
informal views of limit. He selected four students holding each of the two most common
concepts of limit, and one holding each of the other two common concepts.
Each subject was interviewed five times over a period of seven weeks. The first session,
which was an hour long, concentrated on articulating the students' present concept of limit.
The next three sessions, half an hour each, began with students explaining their views of limit
in relation to statements presented by the interviewer, and proceeded to the subjects working
limit problems that brought to the subjects' attention difficulties with their current conceptions
and provided incentive for them to alter their viewpoints. In the final session, which was an
hour long, Williams presented the subjects with each of the three views of limit they had
given in the three previous sessions, and asked them to explain why their views of limit had or
had not changed.
From these interviews, Williams made some generalizations about how the students
viewed limits. He writes:
Students often considered the ease and practicality of a model of limit more important
than mathematical formality. This is particularly true in the sense that models of limit
that allow them to deal with the realities of limits in the classroom, the kind they see
on tests, tend to be seen as sufficient for the purposes of most students. It was noted by
several students that neither formal nor dynamic models of limit figure heavily in the
procedures students use to work problems from their calculus class; their formal
knowledge (e.g., substituting values into continuous functions, factoring and
canceling, using conjugates, employing l'H™pital's rule) is largely separate from their
conceptual knowledge.
The data of this study confirm students' procedural, dynamic view of limit, that is, as
an idealization of evaluating the function at points successively closer to a point of
interest. The data also suggest that there are numerous idiosyncratic variations on this
theme, some of them extremely difficult to dislodge. Given the complex nature of
cognitive change, it is not surprising that the students in the study failed to adopt a
more formal view of limit after only five sessions. . . .
The data suggest that the attitudes toward practicality and mathematical truth
displayed by the subjects did interfere with conceptual change. Specifically, students'
views of mathematical truth, the value they place on practicality and simplicity in
models of limit, the everyday demands of calculus class, their previous experience
20
graphing functions, and their faith in the a priori existence of graphs combine to make
it difficult for them to appreciate the need for a more formal definition of limit. . . .
The end result for the students in this study is a lack of appreciation for formal
thinking, which effectively removed any motivation to learn what is, after all, a very
formal definition of limit. . . .
All this suggests that improving students' understanding of limits from a formal
viewpoint requires careful and explicit instruction, which accounts for the rich variety
of limit models students bring to the classroom as well as the sorts of knowledge they
value. In some sense, their prior knowledge of graphs and functions must be
deconstructed, to expose the underlying assumptions that formal definitions attempt to
address
Robert Davis and Shlomo Vinner (1986) worked with teachers at University High
School in Urbana to develop a mathematics sequence from beginning arithmetic
through two years of calculus, in which concepts were taught first and skills were built
on that understanding. Their paper "The Notion of Limit: Some Seemingly
Unavoidable Misconception Stages" examines the calculus sequence, focusing on the
students' development of the concept of limit. Rather than finding, as they had hoped,
that their method produced greater student understanding, Davis and Vinner found that
changes in instruction did not create as great a change in learning as they had
expected. They write:
“One goal, then, of this present study was to determine how successful this "start with
understanding" approach has been, in the case of the 2-year calculus course. As we
shall see, getting evidence that "early understanding is possible" is difficult, and the
present study can claim little or none. We shall consider why obtaining evidence is so
difficult.” (page 283, italics in the original)
During the first year of the calculus sequence, the students demonstrated mastery of
the limit concept by proving theorems, stating definitions, and producing example
sequences to illustrate weaknesses in incorrect definitions. On the first day of the
following school year, the fifteen students were asked to informally describe the
intuitive concept of the limit of a sequence and give a precise formal definition. The
understanding that they had demonstrated the year before was not always evident in
their responses.
21
Davis and Vinner suggest that many of the students' errors are retrieval errors, similar
in nature to reaching for a phone book and picking up last year's edition instead of this
year's. The students had learned the new concepts, but retained their naive concepts.
When asked about limits, they retrieved the older, naive concepts, and not the newer,
school-taught ones. For instance, many students said that a sequence can never reach
its limit, despite having worked competently the previous year with sequences that did
reach their limits.
These were unusually bright students who had been carefully taught in a conceptually
oriented curriculum intended to prevent them from making the mistakes common to
average students in traditional curricula, but these errors appeared nonetheless. By
examining the students' free responses and considering their ideas in their own words,
this qualitative study was able to find what and how these students think about limits.
It found that they were not much different from less gifted students in traditional
curricula. The same naive concepts were present and affected the students' responses.
Davis and Vinner suggest five explanations for their results: (1) the influence of language; (2)
assembling mathematical representations from pre-mathematical fragments; (3) building
concepts within mathematics; (4) the influence of specific examples; and (5) misinterpreting
one's own experience. The first two have to do with familiar words and phrases, such as
"limit," "approaching," and "going to," and other extra-mathematical ideas that students bring
to their work and use to develop and express their mathematical ideas. These words and ideas
may bring with them misleading connotations. In an effort to prevent confusion, Davis and
Vinner avoided all use of the word "limit" until after the concept was thoroughly developed,
referring instead to the "associated number" of a sequence, but they were not certain that this
was sufficient to prevent students from mixing the mathematical concept of limit with their
ideas of the everyday meaning of the word limit, as in "speed limit" and so on.
Their third explanation also relates to preexisting knowledge, but, unlike the first two, refers
specifically to mathematical knowledge. They write:
Even if words and ideas from outside of mathematics could be excluded from a
student's concept image, and if one could work entirely within mathematics, it would
still be necessary for each student to build mental representations gradually. One
cannot put anything as complex as limit into a single idea that can appear
instantaneously in complete and mature form. Some parts of the idea will get adequate
22
representation before other parts will. It is probably inevitable that these parts will not
be perfectly representative of the whole. Thus there will be stages in the student's
development of mental representations where parts of the representation will be
reasonably adequate and mature, but other parts will not be.
Davis and Vinner designed their instruction to prevent students from making the usual errors
and misconceptions, even going so far as to avoid the use of the word "limit," but found the
same "errors" cropping up despite their efforts. In the passage quoted above, they suggest that
these misconceptions are actually partial conceptions, steps along the road to a full
understanding of limit. The title of their paper characterizes these partial conceptions as
"seemingly unavoidable," suggesting that they are inevitable, and perhaps even necessary, as
students construct an understanding of limit.
This relates to Tall's idea about fundamental obstacles, inherent in the natural ordering of
concepts. One has to build representations gradually, and certain steps in that construction
may come in a necessary order, but that does not mean that the construction will proceed
effortlessly. One cannot come to the study of limits without some preexisting ideas of the
meanings of language and of some mathematical concepts. Indeed, it does not seem desirable
that one should, even if it were possible. Yet, those preexisting ideas, useful though they may
be, also contain obstacles. The needed stepping-stone is at one and the same time a stumbling
block.
Darien Lauten, Karen Graham, and Joan Ferrini-Mundy conducted two one-hour clinical
interviews with each of five calculus students, two taking an advanced placement calculus
course in a high school and three taking first-semester calculus at a state university. They
write:
The purposes of this study were: (a) to examine students' understandings and concept
images of functions and limits; (b) to explore the ways in which their understandings
of these conceptual areas may relate to one another; (c) to attend to the students'
propensity to use of the graphics calculator; and (d) to identify, in a preliminary
fashion, instances where there is evidence that the students' understanding and or
processes for solving problems seem to have been influenced by the available
technology.
This paper reports on interviews with Amy, one of the five subjects, but does not indicate
whether she was a high school student or a college student. The writers mention that Amy had
23
had access to the calculator for only two weeks before the interviews began, and later note
"Amy's unprompted use of the calculator was minimal in the course of these interviews."
Since Lauten, Graham, and Ferrini-Mundy were interested in students' spontaneous use of the
calculator, Amy's lack of unprompted calculator use limited their ability to accomplish parts
(c) and (d) of their purpose.
The interviewer presented Amy with six statements about limits, and asked her to say whether
she agreed or disagreed with each one and to explain why. Amy agreed with all six, but
preferred the first one, "A limit describes how a function moves as x moves toward a certain
point." When presented with example functions graphed on the calculator, Amy used the trace
feature, which allows the user to move the cursor along the curve of the function and read off
the x and y values of each point. Both of these responses are in keeping with the general
observation that Amy appeared to have very dynamic concepts of function and limit. To Amy,
a function is a particle moving along a path, and a limit means the particle is getting close to
some fixed point, but not quite reaching it. Lauten, Graham, and Ferrini-Mundy put it this
way:
In considering Amy's responses to this discussion, she seemed to use "it," "the x value," "the y
value," "the curve," and "the function" interchangeably. In using each of these terms, she
seemed to indicate that they were all on the graph of the function and moving along the curve.
(page 233)
In the case of Amy, the most striking feature of her thinking relative to functions was
her image of the x and y values as particles moving along the graph. We have even
speculated about whether Amy might see the curve as animated in some way, although
our data does not provide enough information to go further with this idea. This
dynamic notion of function may be the basis on which Amy's ideas about limit rest,
because certainly her language in discussing limits includes various motion references.
Lauten, Graham, and Ferrini-Mundy note that Amy seemed to employ different concepts of
function and limit in different contexts, depending on whether the function was represented as
a list of points, a graph, or an algebraic formula. As was the case for ten of the students in Tall
and Vinner's survey of 70 university students, Amy's concept image of limit appears to have
little or no connection to the formal concept definition. It is not clear that she had a concept
definition image at all. Lauten, Graham, and Ferrini-Mundy write:
24
In the limit interview, Amy seemed comfortable with her view that points moved
along a curve and never quite reached the limit point. However, it did not bother her at
all to plug in an x value to get a limit when that was possible in an equation. The
connections between the algebraic manipulation, formal definition, and graphical
interpretation seemed unclear to her. Her expression was "It's all vague anyway." In
fact, the formal definition seemed to have no meaning for her. Her working
definitions, although appearing inconsistent, seemed to satisfy her and serve her well.
There was some evidence that Amy handled equivalent problems quite differently depending
on whether the context was a graphical or an analytical one.
Lauten, Graham, and Ferrini-Mundy present a detailed picture of Amy's conceptions, and
provide several tasks and questions that proved useful in eliciting from Amy responses that
revealed those conceptions. From this, we know a lot about Amy and something about
revealing interview items, but next to nothing about high school or college students as a
population. However, we do have a starting point for further study. As Lauten, Graham, and
Ferrini-Mundy note:
Case studies of this type are especially useful in generating hypotheses and issues to
be explored in subsequent research. . . . For example, given what we learned in the
case of Amy, we have a number of questions about how widespread this tendency to
"trace along the curve" might be in students who have had experience with graphing
calculators.
Since other research has found that most students hold a dynamic view of limit, even when
not using graphing calculators, and since Amy had used the calculator for only two weeks
prior to the interviews, I suspect that Amy's very dynamic concept of limit had little to do
with the curve-tracing capabilities of the calculator. However, there is no way to know this
from what has been reported to date. As Lauten, Graham, and Ferrini-Mundy suggest, their
work sets the stage for further research into that question. The questions and tasks that proved
useful in the study of Amy's conceptions could be adapted to create survey items, which could
then be used to study how widespread Amy's very dynamic concept of limit is, and to what
extent it is related to the use of various technological aids.
Infinity
The concept of infinity but also ideas related to infinitely small and the infinitely large are of
essence in calculus. Kleiner (2001) considers examples of these ideas such as: infinitesimals,
25
indivisibles, differentials, evanescent quantities, moments, infinitely large and infinitely small
magnitudes, infinite sums, power series, limits and hyperreal numbers as they unfolded in the
history of calculus form the 17th through the 20th centuries. He does not limit himself to the
historical account but also present some didactic observations at relevant places.
Tall (2001) talks about natural and formal infinities. He suggests that formal deduction
focuses as far as possible on formal logic in preference to perceptual imagery, developing a
network of formal properties that do not depend on specific embodiments. He shows that
formal theory can lead to structure theorems, whose formal properties may then be reinterpreted as a more subtle form of embodied imagery. He also shows that not only can
natural embodied theory inspire theorems to be proved formally but formal theory can also
feed back into human embodiment.
A number of papers in mathematics education explore students intuitive or tacit knowledge in
regard to infinity prior to students being taught the subject or the influences of this knowledge
once they are taught. Monaghan (2001) for example, explores the views of infinity of young
people prior to instruction. He mainly concentrates on the potential pitfalls of research in the
area of infinity and the work of Piaget, issues concerning the contradictory nature of infinity
and infinity as a process and as an object. He also explores students understanding of infinite
numbers and ideas of infinity in various contexts and tasks. Fischbein (2001) analysed several
examples of tacit influences exerted by mental models on various topics in infinity. He
identified that tacit models which are often uncontrolled consciously, may lead to erroneous
interpretations, paradoxes and contradictions. He dealt with the unconscious effects of figuralpictorial models of statements related to the infinite sets of geometrical points related to the
concepts of function and derivative and to the spatial interpretation of time and motion in
Zeno’s paradoxes. Tsamir, based on such incompatible answers that students provide in
regard to infinite sets discusses an activity which led most of the student to the realization and
the production of contradictory reactions to the same mathematical task and suggests that in
order to avoid such contradiction students need to use the one-to-one correspondence as a
unique criterion for the comparison of infinite sets tasks.
In three sequential papers, a trilogy, Tsamir and Dreyfus describe research-based interviews
which aimed at encouraging an especially talented student to reflect on his thinking about
infinite quantities. The interviews were designed in such a way so that they would use two
incompatible methods to compare the sizes of two infinite sets. The researchers examine
Ben’s reaction to the resulting contradiction. The analysis of the students’ responses is done
26
based on the model of abstraction which is founded on three dynamically nested epistemic
actions of constructing, recognising and building with. In the second paper the authors take
for granted the fact that constructions has occurred in regard to infinite sets and the
researchers offer a theoretical analysis of the consolidation phase. Their analysis showed that
the consolidation phase could be identified by means of the psychological and cognitive
characteristics of self-evidence, immediacy, flexibility and awareness. The third paper in this
trilogy shows that under slight variations of context, knowledge structures that have
apparently been well consolidated may become inactive and subordinate to more primitive
ones.
Derivatives
The articles by Heid and Orton provided the most useful information about student
understanding of the derivative. Heid conducts a study performed on two groups of college
calculus students. One group used computer software extensively in the course while the other
group was taught by more traditional methods. Orton's article focuses exclusively on student
understanding of differentiation in college and pre-college students. His analysis is based on
tasks presented to students in interviews.
Orton shows, based on conceptual tasks given to 110 students majoring in mathematics, that
students have little intuitive understanding as well as some fundamental misconceptions about
the derivative. The routine aspects of differentiation could be performed quite well by almost
all the students interviewed. However, when the students were presented with a function they
had not seen before, the frequency of errors increased indicating strong reliance on
algorithmic steps without a conceptual understanding. Other areas of student difficulty are
related to the tangent as the limit of a set of secants and to the ideas of a rate of change at a
point versus the average rate of change over an interval (Orton, 1983). Heid also notes that
"the notion of derivative as slope or rate of change, or of second derivative as a measure of
concavity, fades quickly with disuse because students learn to rely on memorized procedures
for a small number of exercise types (Heid, 1988)."
Based on the interview tasks assigned to college students training to be teachers of
mathematics, Orton outlined the following items and how well the students performed on the
items. Responses were graded on a five point scale in order to carry out statistical analysis.
Mean scores for each topic range from 0 to 4 and error classification is based on the scheme
described by Donaldson (Orton, 1983). Structural errors are those "which arose from some
27
failure to appreciate the relationships involved in the problem or to grasp some principle
essential to solution." Executive errors were those which involved failure to carry out
manipulations, though the principles involved may have been understood. Arbitrary errors
were said to be those in which the subject behaved arbitrarily and failed to take into account
the constraints laid down in what was given (Orton, 1983).
Table 1. Student understanding of differentiation
Description
gradient of tangent to curve by
Mean
(0-4)
Scores
Error Classification
3.7
Structural and Executive
3.68
Structural and Executive
3.62
Structural and Executive
carrying out differentiation
3.50
Executive
limits of geometric sequences
2.78
Structural
infinite geometric sequences
2.56
Structural
stationary points on a graph
2.54
Structural and Arbitrary
2.02
Structural
average rate of change from curve
1.92
Structural and Executive
use of the -symbolism
1.40
Structural
1.18
Structural
1.14
Structural
differentiation
substitution and increases from
equations
significance of rates of change from
differentiation
rate of change from straight line
graph
rate, average rate, and instantaneous
rate
differentiation as a limit
28
Orton suggests that the basic concepts be revisited throughout the students' mathematical
education. He thinks the initial approach to calculus should be informal and should involve
both numerical and graphical exploration, first using real life data followed by a more
algebraic development of the fundamental ideas (Orton, 1983b, 243-4). Berry & Nyman
(2003) reported on an observational study to show how students’ thinking about the links
between the graph of a derived function and the original function from which it was formed
was enhanced by asking students to “walk” these graphs as if they were displacement-time
graphs. The researchers suggest that these activities help students to extend their
understanding of calculus concepts from a symbolic representation to a graphical
representation and to what they term a “physical feel”.
Integral
Orton's paper entitled, "Students' Understanding of Integration," was the only paper that dealt
exclusively with the topic of integration. Both his differentiation paper and his integration
paper use the same students in an interview format. This paper was the precursor to his paper
on differentiation and contains more background information on the students and the error
classification.
Orton's study on integration shows that students are "able to apply, with some facility, the
basic techniques of integration . . . [but] further probing indicates that they posses
fundamental misunderstanding about the underlying concepts Ferrini-Mundy, J., & Graham
K.G., (1991)." Orton's results, summarized below, indicate that the procedure of breaking up
an area or volume, making use of a limit process, and providing the reasons why such a
method works were not part of the students understanding of the integral.
Table 2. Student understanding of integration.
Description
simplification of sum of
areas of rectangles
heights of rectangles under
graphs
carrying out integration
Mean Scores (0-4)
Error Classification
3.52
Executive
3.42
Structural and Executive
3.40
Structural, Arbitrary and
29
Executive
sequence
approximations
of
to
area 3.22
Executive
under graph
limits of sequences of
numbers
limits from general terms
complications
in
area
calculations
3.06
Structural and Executive
2.90
Structural and Executive
2.78
Structural, Arbitrary and
Executive
limit from sequence of
fractions + from general 2.48
Structural
term
limit of sequence equals
area under graph
volume of revolution
integral of sum equals sum
of integrals
1.00
Structural
0.88
Structural
0.60
Structural
The limiting process, essential in calculus, is generally overlooked before it is suddenly
required. This leads to a great deal of difficulty in this topic. When Orton asked whether it
was possible to obtain an exact answer for the area under the curve y=x2 by taking more and
more rectangles under the curve only 10 students stated that a limiting process was required.
69 students indicated that by taking more and more rectangles under the curve that they could
obtain better and better approximations but that such a procedure would never produce the
correct answer. To address the difficulty in understanding the limiting process Orton suggests
that activities in which the students can explore the idea of a limit in an intuitive way have to
be developed. In these activities the nature of approximations should be emphasized.
30
Tables 1 and 2 show what we can expect in terms of student understanding of differentiation
and integration. Expectations of student performance in various topics should be taken in to
account when developing computer curriculum. Orton suggests the use of computers and
calculators to facilitate a conceptual understanding of the derivative and integral. In particular
he suggests the use of technology to aid in the explorations of the approximation process
stating that "the calculator does provide us with the opportunity of numerical approaches to
calculus, and better understanding of the arithmetic may lead to better understanding of the
algebraically equivalent procedures (Orton, 1983)."
Teaching
In the last two decades there has been an increased interest in the teaching and understanding
of calculus concepts with the latter receiving most attention. In regard to the teaching of
advance mathematical concepts a numbers of researchers (Davis & Hersh, 1981; Dreyfus,
1991; Webber, 2004) suggested that a typical lecture in advance mathematics consists entirely
from definition, theorem and proof. This type of teaching often leads to students’ intimidation
(Kline 1977; Thurston, 1994), gives students a misleading view of the nature of mathematics
(Denis & Confrey, 1996) it hides much of the processes that are used in mathematics
reasoning (Davis & Hersh, 1981; Dreyfus, 1991) and denies students the opportunity to use
their intuition when reasoning about these concepts (Dreyfus, 1991; Fischbein, 1982). The
most important criticism that this approaches gets is that students learn far less than what they
could have learned (Leron & Dubinsky, 1995). A number of researchers have offered
alternative paradigms for teaching advance mathematical courses (Alibert & Thomas, 1991).
Zazkis et al. (2003) describe some pedagogical approaches which aim at resolving students
inappropriate intuitions in regard to function translations.
More recently (Weber, 2004) a lot of attention has been given to the impact of different
teaching approaches of calculus concepts at University level. Weber for example describes the
impact that a traditional definition-theorem-proof format approach has on University students.
Teaching approaches that may enhance students understanding
Cooley (2002) shows the impact that seven formal writing assignments developed for a
calculus I class had on students’ reflective abstraction. The researcher suggested that these
assignments can be an important for the professors to get information in regard to their
students understanding also a good tool for overcoming students’ misconceptions and
facilitating reflective thinking.
31
Moscschkovitch (1994) based on the sociocultural perspective and the concept of
appropriation (Newman, Griffin and Cole, 1989, Rogoff, 1990) describe the way in which a
student learned to work with linear functions. The analysis describes the impact that the
interaction with a tutor had on a learner and how the learner appropriated two aspects of
mathematical practices that are crucial for working with functions: a perspective treating
lines as objects and the action of connecting a line to its corresponding equation in the form
y=mx +b.
Mamona-Downs (2001) discusses how students may be assisted in understanding the limit of
a sequence.
Use of Technology
As indicated by the research described above, students show little intuition about the concepts
and processes of calculus. By mimicking examples and doing homework problems similar to
examples solved in the text students develop misconceptions based on trying to adapt prior
knowledge to new situations (Ferrini-Mundy & Graham, 1991). Students are never given the
opportunity to develop a conceptual knowledge of topics in a traditional calculus class. Heid
shows in her comparison study that students in the traditional class showed no evidence of
attempts to reason from basic principles and lacked detail in defining concepts such as the
derivative. Without a clear conceptual understanding, students base decisions about which
procedure to apply on the given symbols and ignore the meaning behind the symbols (White
& Mitchelmore, 1996).
To remedy this, Thompson suggests that the curriculum should
be problem based, promote reflective abstraction; contain . . . questions that focus on
relationships; have as its objective a cognitive structure that allows one to think with
the structure of the subject matter; and allow students to generate feedback from
which they can judge the efficacy of their methods of thinking (Thompson, 1985).
One area being discussed in mathematics reform is the use of technology in the calculus
course. Many topics in mathematics have characteristics which indicate that a computer aided
learning environment may be effective in promoting student understanding.
[Topics in mathematics which lend themselves to computer implementation] have
visual aspects which can be well represented on a computer screen; they have
transformational aspects which necessitate a dynamic implementation; they have
technical computational aspects which are not very relevant to the essence of the topic
32
and are thus better being taken care of by the computer; and they are intimately
connected to the relationship between two different representations of the same
concept; these two representations can be dealt with in parallel by the computer
program (Dreyfus & Halevi, 1990/91).
Using the computer as a tool for performing the procedures of calculus and algebra can free
students to explore applications (Hsaio, 1984/85; Tall, 1986). The course can then deemphasize skills and concentrate on the underlying concepts. Students in the study done by
Heid stated that they enjoyed the computer work because it freed them from the manipulative
work and gave them confidence in results which were based on their reasoning. It also
allowed them to focus more attention on the global aspects of problem solving. Heid's study
showed that students using the computers understood the concepts as well as, and in most
cases better than the students in the comparison class. After only three weeks of work on
traditional skills the experimental class performed almost as well on the final examination (a
skills test) as the comparison class which met 200 minutes a week for 15 weeks.
Research in education must be considered when designing computer software. Kaput and
Thompson state that to use "technology . . . one must continually rethink pedagogical and
curricular motives and contexts. To exploit the real power of the technology is to transgress
most
of
the
boundaries
of
the
school
mathematics
practice
. . . Normally a powerful technology quickly outruns the activity-boundaries of its initial
design (Kaput & Thompson, 1994)." Often such technology-based tools are "designed by
people steeped in the technology but without deep insight into the problems of mathematics
education (Kaput & Thompson, 1994)." People developing software and hardware should
therefore work to discover something about the learning process as it occurs with the support
of that software or hardware (Ferrini-Mundy & Graham, 1991). Damarin and White offer the
following characteristics for high quality educational software:
1. Appropriate: The program should preserve the integrity of the subject matter and
respect the integrity of the learner. The instructional goals of the program should be
appropriate to the intended user and the format of the presentation should be designed
to incorporate appropriate learning theory.
2. Friendly: The user should be able to interact easily and naturally with the software
with a minimum of confusion.
33
3. Simple: The structure of the program should be as clear-cut and direct as possible.
Rules for using the software should not be complicated.
4. Flexible: The software should lend itself to use in a variety of related learning
situations. The software should be adaptable to the varying needs of teachers and
learners.
5. Robust: The software should be designed to accept unusual responses and be able to
process them in a manner meaningful to the user.
6. Constructible: The topic selected for development must be such that a meaningful
instructional program can be designed within the limitations of the available hardware
and software tools.
7. Verifiable: The software, embodying the concept to be taught, must correctly model
the computer experience planned by the designers.
8. Parsimonious: The software should make effective use of the computer capabilities
available (Damarin & White, 1986).
Computer software must also be aware of topics which have proved to be difficult for
students in a calculus course. Based on the research in student understanding, in the list
below, we suggest topics and difficulties that must be addressed.
In the topic of functions and variables students were found to:

view a function as a single formula

have trouble viewing functions in an across-time manner

not see the connection between algebraic and graphical representations of a function

treat variables simply as symbols to be manipulated
In the topic of limits and continuity the research showed that:

often students' conceptual understanding is based on colloquial meaning of terms
involved

students often focus on the getting close to rather than on the number being
approached

students often identify the notion of a limit with a process rather than a number
When dealing with derivatives students have difficulty with the following items:
34

rate of change from straight line graph

average rate of change from curve

use of the -symbolism

rate, average rate, and instantaneous rate

differentiation as a limit
And when learning integration the following items proved difficult:

limit of sequence equals area under graph

volume of revolution

integral of sum equals sum of integrals
35
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