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representation as a ‘topic’ to be learned in isolation. Even when students have various links
between different representations of functions, overall these links may not imply the core
concept of function. Therefore, investigating students’ understanding of the function
concept by focusing on the notion of core concept of function becomes more crucial.
There are various aspects of a function as given below:
• Formal set of ordered pair definition*
• A colloquial definition (in everyday language)*
• A function box (input-output box)
• A set of ordered pairs (considered set-theoretically)*
• A set diagram (two sets and arrows between them)*
• A table of values
• Graphs (drawn by hand or computer)*
• Expressions*
Starred aspects are the focus of attention in the Turkish context. Therefore, these will be
focused as the data in this study.
4.3 Simplicity and complexity of the core concept of function
Mathematically, the core concept of function has both its simplicity and complexity. The
words “simplicity” or “simple” will be used in a particular way that may be different from
their use in everyday language. It is simple in the sense that the properties of it are
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element in the second set. It is complex in the sense that it has a richness and it gives
access to a variety of ideas (Akkoç & Tall, 2002). In other words, it acts as a unifying
concept for different mathematical ideas. Some students focus on the essential of the
concept definition which is central to the wider complexity. However, for many other
students, the function concept may continue to be cognitively complicated in the sense that
poorly connected ideas continue to persist without being coherently linked. For instance,
some students focus on the details in different contexts, therefore could not overcome the
complications.
4.4 Different aspects of functions: Prototypes versus exemplars of functions
Mathematically, the function concept belongs to a clear-cut category, the category of
function. Something is either a function or not. However, for students some aspects are
better examples of functions and these better examples are different for each student.
Therefore, the category of function might be fuzzy for a student. As discussed in the
literature review chapter, a category can be represented by prototypes or by exemplars,
prototypes representing general ideas and exemplars as more specific cases. To explore the
complications of the function concept, the theoretical framework makes a distinction
between prototypes and exemplars.
The fundamental finding from the preliminary study was that the core concept of function
is not the focus of attention for most of the students when dealing with different aspects of
functions. Students had much more difficulty with graphs and expressions compared to
set-correspondence diagrams and sets of ordered pairs. The personal concept definition of
a student is more likely to be ‘informally operable’ for set-correspondence diagrams and
sets of ordered pairs than graphs and expressions. This finding was explained by the fact
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expressions. Instead of using the definitional properties, students rely on prototypical
examples of graphs and expressions. However, when dealing with set diagrams and sets of
ordered pairs, they do not develop prototypical examples, but are more focused on the core
concept of function by using the definitional properties. Therefore, in the preliminary
study, a distinction between using the definitional properties and relying on prototypical
examples was made. Considering these results, the theoretical framework of this study
focuses on this distinction from a different point of view. Instead of using the term
“prototypical example”, the term “exemplar” is used to emphasize that graphs and
expressions are more specific cases (Akkoç & Tall, 2002).
Two variations of Prototype=Representation interpretation of prototype effects for
categorization are chosen to be a starting point to distinguish between set-correspondence
diagrams and sets of ordered pairs on the one hand, and graphs and expressions on the
other. As discussed in the literature review in section 2.2.2, one variation of
Prototype=Representation interpretation suggests that prototype is an abstraction, say a
schema or a feature bundle. A second variation suggests that the prototype is an exemplar,
that is, a particular example (Lakoff, 1987b). Instead of using the term ‘prototypical
examples’ for graphs and expressions as in the preliminary study, function graphs and
expressions are treated as exemplars, as more specific cases. The aim of making a
distinction between prototypes and exemplars is not to claim that human beings categorize
by developing prototypes or exemplars. The reason for starting with a theoretical
framework which makes such a distinction is that some aspects of functions are taught in a
prototypical way while some aspects (such as graphs and expressions) are taught in
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linear, polynomial, logarithmic, exponential functions. These are taught as different topics
at different stages in the curriculum.
4.5 Core concept of function and prototype – exemplar distinction
So what is the relationship between the core concept of function and prototype and
exemplar distinction? As discussed in the preliminary study, students who could not use
their personal concept definitions, heavily relied on the prototypical examples. As
discussed in Akkoç & Tall (2002), students’ responses revealed a spectrum of performance
ranging from students who have strong focuses on the core concept of function to students
who can hardly refer to definitional properties for different aspects of functions.
Furthermore, students’ responses to prototypical aspects of functions differed from their
responses to exemplars. Exemplars of functions caused more complications.
The theoretical framework of this study considers coherency in recognizing different
aspects of functions (both prototypes and exemplars) correctly with a strong focus on the
definitional properties as an indication of the ability of focusing on the core concept of
function as a cognitive unit. Therefore in the analysis this coherency is considered to
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