Content uploaded by Vesife Hatisaru
Author content
All content in this area was uploaded by Vesife Hatisaru on Mar 08, 2022
Content may be subject to copyright.
For Peer Review Only
Investigating Secondary Mathematics Teachers’ Analogies
to Function
Journal:
International Journal of Mathematical
Education in Science and Technology
Manuscript ID
TMES-2021-0100.R1
Manuscript Type:
Paper
Keywords:
analogy, mathematical knowledge for
teaching, structure-mapping, the concept
of function
<a
href="http://www.ams.org/mathscinet/msc/msc2020.html"
target="_blank">2020 Mathematics Subject
Classification</a>:
Functions, Mathematics education
URL: http://mc.manuscriptcentral.com/tmes Email: tmes-peerreview@journals.tandf.co.uk
International Journal of Mathematical Education in Science and Technology
For Peer Review Only
1
Investigating Secondary Mathematics Teachers’ Analogies to Function
This study investigates the analogies used by a sample of secondary
mathematics teachers when they described the concept of function. The study is
concerned with understanding what analogies were used by the participant
teachers and conceptions of functions positioned in those analogies. Using
examples from a set of responses to five open-ended questionnaire items, the
article presents the main analogies found in the descriptions of the participant
teachers and examines ways in which these are structurally mapped to function
by applying Gentner’s structure-mapping theory of analogy. Among those
discussed are the child-mother linkage and machine/factory analogies. Of
interest is the predominance of illustrations that show a correspondence
approach to functions (mappings and input-output machines), while a
covariation approach is entirely absent. The article makes the case for using the
concept of analogy as a tool in research on teachers’ Mathematical Knowledge
for Teaching (MKT).
Keywords: analogy; mathematical knowledge for teaching; structure-mapping;
the concept of function
Introduction
The concept of function is a challenging but critical mathematical concept for secondary
school students (Son & Hu, 2015). Over the past four decades, a considerable body of
research has been focused on high school or college students’ understanding of functions
(e.g., Ayalon et al., 2017; Author et al., 2017; Clement, 2001; Dubinsky & Wilson, 2013). In
many cases, students’ concept images of function have been found to be limited. In Clement
(2001), only four out of 35 high school students were able to provide a definition of function,
which requires that every element in the domain must be mapped to exactly one element in
the range. Most students perceived function as either an image of a machine or an image of a
graph that passes the vertical line test. In Author et al. (2013), among 130 grade ten students
studying at a vocational high school, only thirty students’ descriptions met the main
mathematical features of a function to some extent, while many other statements (n=79)
included no mathematically semantic associations of function but provided colloquial
sentences involving the word ‘function’ (e.g., functions of a mobile phone).
Page 1 of 26
URL: http://mc.manuscriptcentral.com/tmes Email: tmes-peerreview@journals.tandf.co.uk
International Journal of Mathematical Education in Science and Technology
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
For Peer Review Only
2
Investigating Secondary Mathematics Teachers’ Analogies to Function
How could students’ understanding of functions be improved? As students’
performance in functions is influenced by several factors including textbooks and curriculum
(e.g., Ayalon et al., 2017; Son & Hu, 2015), the understanding of function cannot be only or
directly linked to the quality of the teachers’ knowledge of the relevant subject. Nevertheless,
the mathematics education community generally emphasises that teachers’ knowledge,
especially their Mathematical Knowledge for Teaching (MKT) (Ball et al., 2008), plays a key
role in students’ learning of this content (Author et al., 2017; Stein et al., 1990). The analysis
of teachers’ understanding of and approaches to function could provide insights into
influences on students’ learning of function. As part of a larger qualitative exploratory study
investigating secondary mathematics teachers’ MKT about the concept of function, with a
focus on the Common Content Knowledge (CCK) and Specialized Content Knowledge
(SCK) domains of MKT (Author, 2020), this paper explores the analogies used by teachers
when they discuss functions, and examines ways in which teachers’ analogies are structurally
mapped to function. Analogy is defined as the similarity between different objects (e.g., the
concept of function and a function machine), both of which hold the same system of relations
(Gentner & Maravilla, 2018). The paper is an extension of work originally presented in a
British Society for Research into Learning Mathematics (BSRLM) conference in November
2021, and an early version of it was published in the conference proceeding (Author, 2021)
Rationale for the Study
This study addresses several identified gaps in research on the teaching and learning of
function. First, students’ (mis)understanding of function is a critical concern in mathematics
education. Students tend to have expectations of function that have no logical grounding and
do not relate to the definition of function, such as not accepting unfamiliar relations (e.g.,
constant, or piecewise relations) as functions (Clement, 2001; Author et al., 2013; Oehrtman
et al., 2008). To support students to develop a robust understanding of function and avoid
common misconceptions (Watson & Harel, 2013), teachers need to have a deep knowledge of
functions-related content (Author et al., 2017; Nyikahadzoyi, 2015; Roux et al., 2015) and
the knowledge and skills of this content specific to its teaching (Steele et al., 2013).
Teachers’ knowledge is also required to be able to enrich students’ experiences beyond
curriculum and textbook expectations, so students might know more than the typical
curriculum and textbook descriptions (Ayalon et al., 2017; Author et al., 2017). Exploring
Page 2 of 26
URL: http://mc.manuscriptcentral.com/tmes Email: tmes-peerreview@journals.tandf.co.uk
International Journal of Mathematical Education in Science and Technology
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
For Peer Review Only
3
Investigating Secondary Mathematics Teachers’ Analogies to Function
teachers’ analogies can provide insight to teachers’ understanding of function and give access
into the learning opportunities students receive in the teaching of functions.
Second, in teaching, one of the central focuses is on the presentation of concepts to
students in the form of explanations, demonstrations, examples, and analogies (Rowland,
2013; Shulman, 1986). Teachers usually preface their instructions with terms such as it is
like, similarly, likewise, or just as. In these instances, teachers generate or use analogies to
explain concepts to students (they may sometimes even be unaware of using analogies)
(Glynn, 2008). Analogies are, therefore, attractive in teaching as they are simple ways to
explain abstract scientific concepts in familiar terms (Aubusson et al., 2006; James &
Scharmann, 2007) to communicate those concepts effectively to students of different
backgrounds and prior knowledge. They are “vehicle[s] of thought” which can enhance
mathematical learning and understanding (English, 1997, p. 4) and may be a form in which
the MKT is held (Hill et al., 2008). Recently, making analogies has been identified as a way
of establishing mathematical connections (Rodríguez-Nieto et al., 2020). However, the
research in this field is limited to studies examining pre-service teachers’ analogies to
function (Espinoza-Vásquez et al., 2017; Ubuz et al., 2009) and not necessarily incorporating
the critical theoretical ingredients of analogies into investigations. These efforts may fail to
utilise the power of analogies as a way of understanding or enhancing teachers’ knowledge
(see McCulloch et al., 2020). The study reported in this paper builds upon earlier research by
providing a comprehensive portrayal of the analogies secondary mathematics teachers made
when discussing function, applying structure-mapping in analogy (Gentner, 1983, 1989;
Gentner & Markman, 1997).
Third, school mathematics curricula and textbooks at school or college level present
different definitions of function (Cooney et al., 2010; Thompson & Carlson, 2017), and many
of those definitions represent a correspondence approach to understanding functions.
Considering the increasing emphasis on inquiry, modelling, and the integration of STEM
disciplines to many curriculum documents (e.g., English, 2015), students need to develop a
strong understanding of functions and functional perspective in different real-world contexts.
Students should be provided with opportunities to investigate the relationships between
quantities and variables whose values vary or covary (Thompson & Carlson, 2017). The way
teachers teach functions is chiefly related to what they know and value, and what they
consider important for their students to know about functions (Cooney & Wilson, 1993). It is
Page 3 of 26
URL: http://mc.manuscriptcentral.com/tmes Email: tmes-peerreview@journals.tandf.co.uk
International Journal of Mathematical Education in Science and Technology
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
For Peer Review Only
4
Investigating Secondary Mathematics Teachers’ Analogies to Function
important to investigate teachers’ orientations towards functions and access the (dominant)
views of function that they possibly emphasise in their teaching.
Research Questions
This paper explores the occurrence of concrete, real-world manifestation of functions
revealed in the descriptions of the teachers. In particular, a sample of Turkish secondary
mathematics teachers was prompted to comment on imagined student responses and teaching
situations through an open-ended questionnaire on the concept of function. The paper aims to
identify the analogies used in the responses of the teachers and examine ways in which they
are structurally mapped to function, and also describe the conceptions of function revealed in
those analogies. Data analysis focuses on any relational comparison that arose in the
participant teachers’ responses, such as functions as machines. The research questions that
guided the study are: (1) What analogies are evident from the descriptions of teachers? and
(2) What views of function do those analogies illustrate?
In Authors et al. (2017) and Author (2020), we reported an extensive review of MKT
framework and secondary mathematics teachers’ MKT about functions. Relevant to this
article, in the following sections I present the two characteristics of functions that distinguish
them from other mathematical relations, different conceptions of function, and the theoretical
framework used, i.e. structure-mapping in analogy. I then describe function analogies in the
research literature before introducing the study methods.
Background Literature
Essential Understanding of Function
A function is a special kind of mathematical relation (Thorpe, 1999) “that uniquely associates
members of one set with members of another set” (Stover & Weisstein, 2017, para. 1). Stover
and Weisstein (2017) writes that “More formally, a function from A to B is an object f such
that every a ϵ A is uniquely associated with an object f (a) ϵ B” (para. 1). The features of
uniqueness (also known as single-valuedness) and arbitrariness distinguish function from
other mathematical relations (Even, 1990; Cooney et al., 2010). The requirement of
uniqueness is in the relationship between the two sets (domain and range) on which the
function is defined; each element of the domain maps exactly one element of the range. The
uniqueness feature, therefore, does not allow one-to-many relations, it only allows one-to-one
Page 4 of 26
URL: http://mc.manuscriptcentral.com/tmes Email: tmes-peerreview@journals.tandf.co.uk
International Journal of Mathematical Education in Science and Technology
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
For Peer Review Only
5
Investigating Secondary Mathematics Teachers’ Analogies to Function
and many-to-one relations. The requirement of arbitrariness is that a function need not be
defined by any particular sets of objects; nor do the sets even need to be numbers, and even if
a function is defined this way, it need not demonstrate regularity, nor does it need a particular
graph or expression to define it (Even, 1993). While uniqueness is explicit in definitions of
function, arbitrariness may be invisible as a feature in function definitions (Steele et al.,
2013). Nevertheless, any mathematically valid conception of function should certainly
include both univalence and arbitrariness, and these features of function should be explicitly
addressed in the teaching and learning of functions (Cooney et al., 2010). As these features
can be one of students’ main difficulties in understanding function, they need to be unpacked
while teaching functions (Tabach & Nachlieli, 2015).
School mathematics curricula and textbooks at school or college level present
different definitions of function (Cooney et al., 2010; Thompson & Carlson, 2017), and many
of those definitions represent a correspondence approach to understanding functions (see
Table 1). The correspondence approach can facilitate the notions of domain and range and
contribute towards understanding that relations between two sets of numbers might be shown
as algebraic rules by which input values get transformed into output values (Ayalon et al.,
2017). A covariation approach is about understanding how variables change together, so that
a change in one variable relates to a change in another (Ayalon et al., 2017). Concepts such
as variable, rate of change, and the notion of covariation, in which we reason about the
simultaneous varying of the values of two or more quantities, or continuous covariational
reasoning, are central to an understanding of calculus and to the modelling of dynamically
changing phenomena in science and engineering (Thompson & Carlson, 2017).
Table 1: Approaches to function, taking into account different views of function.
Approach to
function
View of function
A function is a:
Correspondence
taking inputs to outputs
manipulation or operation
a mapping from one set to
another
correspondence/mapping between the elements
of two sets
a rule assigning x to f(x)
rule
a set of ordered pairs
relation and/or a Cartesian product, a set of
ordered pairs such that no two pairs have the
same first entry but different second entries
Covariation
quantities varying
simultaneously
change in one variable that is related to a
change in another, or how the variables change
together
Note. The table is created based on Ayalon et al. (2017) and Cooney et al. (2010)
Page 5 of 26
URL: http://mc.manuscriptcentral.com/tmes Email: tmes-peerreview@journals.tandf.co.uk
International Journal of Mathematical Education in Science and Technology
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
For Peer Review Only
6
Investigating Secondary Mathematics Teachers’ Analogies to Function
Ideas of variation and covariation are, therefore, considered necessary for students in order to
build concepts of function that are both useful and reliable, contributing fundamentally to
their mathematical development (Thompson & Carlson, 2017).
Teachers’ Understanding of Function
The teaching and learning of functions are well-researched areas in mathematics education. A
considerable body of research exists which focuses on high school or college students’
(mis)understanding of functions (e.g., Ayalon et al., 2017; Clement, 2001; Dubinsky &
Wilson, 2013; Author et al., 2017) and pre-service or practicing teachers’ (in)capacity to
teach functions (e.g., Author et al., 2017; McCulloch et al., 2020; Tabach & Nachlieli, 2015).
The nature of teacher knowledge that is required for effective teaching of functions (e.g.,
Author, 2020; Nyikahadzoyi, 2015; Steele et al., 2013; Wilkie, 2014) and its influence on
student learning (Author et al., 2017; Watson & Harel, 2013) have also been explored. It is
suggested that teachers of mathematics should know and understand the uniqueness and
arbitrariness characteristics of functions (Cooney et al., 2010; Even, 1990; Nyikahadzoyi,
2015; Steele et al., 2013) and possess a wide-ranging repertoire of examples that best
illustrate them (Nyikahadzoyi, 2015). Teachers should hold a strong covariation perspective
of functions, as well as the correspondence perspective (Thompson & Carlson, 2017).
Research studies, nevertheless, show that both pre-service and practicing teachers
sometimes have incomplete and inconsistent understandings of content and pedagogical
content knowledge related to function. These involve, for instance, relying on internal
representations of function rather than the formal definition of function, and thinking that a
function can or must always be represented by an equation, or expecting functions to be
defined by numbers only, or considering familiar graphs (circles, ellipses) as functions (e.g.,
Cooney, 1999; Even, 1993; Even & Tirosh, 1995; Hitt, 1998; Stein et al., 1990) (for
comprehensive reviews see for example Author, 2020 and Zazkis and Marmur, 2018).
Designing teacher professional learning around classroom practices (Ribeiro & Ponte, 2019)
or using a machine analogy as a cognitive root for function (McCulloch et al., 2020) have
been found promising in terms of improving teachers’ knowledge regarding the concept of
function, which in turn affects student learning (Author et al., 2017).
Page 6 of 26
URL: http://mc.manuscriptcentral.com/tmes Email: tmes-peerreview@journals.tandf.co.uk
International Journal of Mathematical Education in Science and Technology
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
For Peer Review Only
7
Investigating Secondary Mathematics Teachers’ Analogies to Function
Conceptual Background
Structure-mapping Theory on Analogy
An analogy is a mapping of knowledge from one domain (the base) into another (the target)
and conveys two domains that share a relational structure independently of the objects in
which the relations are embedded (Gentner, 1989). The structure-mapping theory posits that
such commonalities convey that the attributes held by the base objects also held by the target
objects (Gentner et al., 2001). For example, the anger is like a tea kettle analogy (Gentner &
Maravilla, 2018) defines a mapping from the object of tea kettle to the object of anger. As it
is the domain being explicated, anger is called the target. As it is the domain that serves as
the source knowledge, tea kettle is called the base. Assume that the illustration of the base
domain (tea kettle) is stated in terms of object nodes such as burn and predicated as hot tea
kettle can burn the skin, and the target domain (anger) has object nodes such as destroy,
damage and is predicated anger can destroy relationships. The analogy maps the object nodes
of tea kettle onto the object nodes of anger. The objects that make up the two domains are
different (tea kettle and anger), but it is the common relations that are essential to the
analogy, not the common objects (Gentner, 1989). The essence of the analogy between the
tea kettle and anger is, for example, that both can be destructive. Through connecting the
abstract concept (anger) with a more familiar, concrete concept (tea kettle), the analogy
serves understanding of the abstract, complex target concept (Glynn, 2008). This
characteristic of analogy makes it a useful cognitive tool (Gentner & Markman, 1997).
In conventional metaphor systems which can be considered as a species of analogy
(Bowdle & Gentner, 2005), two kinds of metaphors are defined: grounding and linking
metaphors. Grounding metaphors represent directly grounded ideas and allow you to project
from everyday experiences onto abstract concepts, for example subtraction as taking objects
away from a collection. Linking metaphors present abstract ideas, such as geometrical figures
as algebraic equations (Lakoff & Nuñez, 2000). The same concept applies to structure-
mapping. That is, between analogy and abstraction there is a continuum. In both analogy and
abstraction preliminary relations are matched, and in both cases a relational structure is
mapped from base to target. If the base representation includes concrete objects whose
individual attributes must be left behind in the mapping, the comparison is analogy (heat is
like water). As the object nodes of the base domain become more abstract, the comparison
Page 7 of 26
URL: http://mc.manuscriptcentral.com/tmes Email: tmes-peerreview@journals.tandf.co.uk
International Journal of Mathematical Education in Science and Technology
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
For Peer Review Only
8
Investigating Secondary Mathematics Teachers’ Analogies to Function
becomes abstraction (heat flow is a through-variable) (Gentner et al., 2001). This article is
devoted to analogy which coincides with Lakoff and Nuñez’s (2000) grounding metaphors.
As noted earlier, analogies are comparisons that share chiefly relational
commonalities (Gentner, 1983). Nevertheless, sometimes comparisons are made based only
on surface features, especially when concepts are less well understood (Richland &
McDonough, 2010). In the structure-mapping framework, this type of comparison, where no
significant attributes or relational commonalities are shared, is called anomaly. Lakoff and
Nuñez (2000) call these extraneous metaphors. In the case of mathematics, extraneous
metaphors or anomalies are not connected at all with the structure or fundamentals of
mathematics. For example, the idea of step function can be drawn to look like a staircase.
Although the staircase image can be useful for visualisation, it has nothing to do with the
fundamentals or the actual content of the mathematics (Lakoff and Nuñez, 2000). Another
example of anomaly or extraneous metaphor in mathematics is the word function. In
everyday language, the word function has multiple meanings including purpose, feature, and
functionality. Yet, it has specific mathematical meanings. As the word is defined and used
differently in everyday life and in mathematics, each use originates a different discourse
(Tabach & Nachlieli, 2015). Although these meanings can be used to support students to
move into the technical, mathematical definitions (Meiers & Trevitt, 2010), function’s
everyday meanings differ considerably from its mathematical meanings.
In interpreting a scientific analogy, the objects of the base domain are mapped to the
objects of the target domain to obtain the maximum structural match (Gentner, 1989). The
strength of an analogical correspondence depends, therefore, on the overall degree of
structural overlap (Gentner, 1983).
Analogy in Functions
Author (2021) summarises that students do not necessarily have the background knowledge
to learn unfamiliar or abstract mathematical concepts such as the concept of function. For
teachers, one of the useful ways to assist students’ learning is to establish links between the
new or abstract concepts and the concepts students are already familiar with or the
knowledge which they already have. An analogy can be used for this purpose – to make
unfamiliar concepts familiar in the learning process (Duit, 1991). For example, the widely
used function is (like) a machine analogy by teachers or in textbooks (e.g., Espinoza-Vásquez
Page 8 of 26
URL: http://mc.manuscriptcentral.com/tmes Email: tmes-peerreview@journals.tandf.co.uk
International Journal of Mathematical Education in Science and Technology
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
For Peer Review Only
9
Investigating Secondary Mathematics Teachers’ Analogies to Function
et al., 2017; Ubuz et al., 2009; Unver, 2009; McCulloch et al., 2020; Willoughby, 1999)
defines a mapping from the object of machine to function. Author (2021) writes, in this
analogy, function is the explicated object and called the target, whereas machine is the
concept that serves as the source of knowledge and called the base. In the analogy, the
illustration of the base domain (machine) is stated in terms of object nodes such as transform,
and the target domain (function) has object nodes such as evaluate, change. The analogy
maps the object nodes of machine onto the object nodes of function and predicates that a
function evaluates or changes objects just as a machine transforms things. Both of these
objects receive some input and give an output, and this is what the essence of the analogy
between them.
Sand (1999) proposed a concrete, real-life example that could be used to explain
concepts related to functions, the mail carrier analogy. Sand aimed to give his students a
general notion of functions based on a comparison between what a mail carrier does and
functions. By comparing functions with a mail carrier, we can infer that the domain and range
sets of a function can be something other than numbers (i.e. letters and mailboxes,
respectively), and this promotes understanding of arbitrariness within functions. The
relational similarities between these two different situations are as follows: each domain
value is mapped to exactly one range value, i.e. each letter is placed exactly in one mailbox.
Every domain value is mapped to range values, every letter is delivered, and one domain
value cannot be mapped to more than one range value, one letter cannot be placed in two
mailboxes (uniqueness). Several domain values may be mapped to one range value, several
letters may be sent to one mailbox (many-to-one mapping), and some range values may
remain unmapped, some mailboxes might remain empty (upper bound for the range). The
specific types of functions that could be illustrated through this analogy include: a constant
function, all letters are delivered in the same mailbox; a one-to-one function, each mailbox
receives no more than one letter; and an onto function, no mailboxes remain empty.
Davidenko (1999) gave the examples of asking students to say the name of their
classmates as they point to them on the class picture, or to say what the prices are at the
school store (notebooks are $10, pencils are $2, and erasers are $1). In the former situation,
domain is the set of students in the picture; range is the set of names of students; and the
function is f (picture) = name. For example, f (boy wearing glasses) = Jeremy. A response to
a question such as Why was Sari not mentioned? could be, Sari is not in our class, which
indicates an element not in the domain of the function. In the latter example, domain is the set
Page 9 of 26
URL: http://mc.manuscriptcentral.com/tmes Email: tmes-peerreview@journals.tandf.co.uk
International Journal of Mathematical Education in Science and Technology
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
For Peer Review Only
10
Investigating Secondary Mathematics Teachers’ Analogies to Function
of products available at the school store, range is the set of prices, and the function is defined
as g (product) = price. An example, g (eraser) = $1. If students were asked if there is any
product that costs $15, they might respond that nothing in the store costs more than $10,
suggesting an upper bound for the range of the function (see Davidenko, 1999).
The analogies mentioned take a correspondence approach to building the concept of
function and can develop a rule of correspondence view (e.g., by looking across rows
vertically in domain and range sets to search for an invariant relationship between entries) in
students. However, they do not address how students can think about what happens between
entries in domain and range sets (i.e. covariation). Situations conveying the covariation
perspective can include, for example, when a student “conceives of a runner’s distance from
a reference point as varying and of the elapsed time measured on a stopwatch as varying.
Uniting the two in thought so that they vary simultaneously constitutes covariational
reasoning” (Thompson & Carlson, 2017, p. 426).
The Study
Context
As mentioned earlier, this study is a qualitative exploratory study investigating secondary
mathematics teachers’ MKT about the concept of function. A sample of Turkish secondary
mathematics teachers was prompted to comment on imagined student responses through an
open-ended questionnaire on function (Author, 2020). As part of the larger study, the current
paper examines the analogies used in the descriptions of the teachers and describes the
conceptions of function revealed in those analogies.
As a country with a centralized school system, the Turkish school system is regulated
by the Ministry of National Education (MoNE). The school curriculum is designed by the
Board of Education, and textbooks are officially approved by the Board. Students are
formally introduced to functions in secondary schools at grade 9 (age 15-16 years). At the
time of the study, in the secondary curriculum (Board of Education, 2011), the concept of
function was primarily based on the set-theoretic, correspondence perspective. The concept
was presented through an activity which maps a group of children (domain with the houses in
a neighbourhood (range), emphasizing that each child lives in a house (uniqueness) and there
might be houses that are not associated with children (upper bound for the range). The
relation between the concept of Cartesian product, relation and function was emphasized. The
Page 10 of 26
URL: http://mc.manuscriptcentral.com/tmes Email: tmes-peerreview@journals.tandf.co.uk
International Journal of Mathematical Education in Science and Technology
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
For Peer Review Only
11
Investigating Secondary Mathematics Teachers’ Analogies to Function
learning objectives for each concept were itemized (e.g., Explain what a relation is, represent
it through a diagram and draw its graph) followed by hints and suggestions for explanations,
including key examples or exercises.
In the accompanied grade 9 textbook (MoNE, 2012), two warm-up activities were
utilized in the introduction to functions. The first activity included various visuals depicting
that a function is a mechanism converting inputs to outputs (e.g., students start school, study,
and graduate). The second activity depicted various mappings between the set of customers at
a restaurant (domain) and the list of dishes (range) that they can order according to two
conditions: all customers will have a meal, and each customer can order only one dish
(uniqueness). By means of this activity, it was highlighted that a function is a special relation
which maps the elements of two sets and does not allow one-to-many relations, i.e. no one
can order more than one dish (for more details, see Author et al., 2017). The covariation
perspective of functions was less dominant in the grade 9 textbook. Basic function concepts
were generally presented with functions defined in infinite sets, and then the focus solely
shifted to real numbers or infinite intervals, which were the immediate subsets of real
numbers. The textbook involved no learning tasks based on real-world situations examining
functional relations to highlight the uses of functions in different contexts. In the curriculum,
it was not until grade 12 that students were introduced to piecewise functions thorough an
example involving the pricing policy of a shipping company that applies fixed price ranges
for items in certain weight intervals. This approach to functions has persisted in the updated
textbook (MoNE, 2019).
Data and Analysis
Data for this paper came from Author (2020) investigating secondary mathematics teachers’
MKT about function, with a focus on content knowledge. The paper focuses on the analogies
identified in the responses of 42 teachers (31 female and 11 male) who voluntarily
participated in the study. The teachers taught at fifteen different vocational high schools in
the capital city of Turkey, Ankara. The average teaching experience of all teachers was 13.5
years, ranging from two years to 25 years. Twenty-four of the participants held a bachelor’s
degree in mathematics and 17 in mathematics education. None of the teachers had
participated in any sort of professional learning activities specific to the teaching and learning
of functions.
Page 11 of 26
URL: http://mc.manuscriptcentral.com/tmes Email: tmes-peerreview@journals.tandf.co.uk
International Journal of Mathematical Education in Science and Technology
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
For Peer Review Only
12
Investigating Secondary Mathematics Teachers’ Analogies to Function
The data were collected through the Function Concept questionnaire, with the
approval of the relevant university ethics committee and the Ministry of National Education
(MoNE). To increase the participation rate and allow participants to think and reflect about
the items without any time restriction, the teachers completed the questionnaire during their
free time. The questionnaire sheets were collected during my personal visits, or they were
mailed to me by the teachers in a sealed envelope to protect participant confidentiality. The
teachers were designated codes (T1, T2, and so on).
The Function Concept questionnaire includes twelve items. Assuming that teachers’
conceptions and knowledge are situated in practice (Brown et al., 1989), the questionnaire
items prompted teachers to discuss on anticipated student questions and teaching situations.
Relevant to this article are five items focusing on anticipated, desired, and assessing
understanding type of responses (see Table 2) (for the other items see Author, 2020). Items 1
and 2 (anticipated) aimed to having the teachers envision how students might mathematically
approach the concept of function. This involved the teachers’ expectations about how
students might define and exemplify function in their own words, the array of students’ both
complete and incomplete definitions and exemplifications. Assuming to get access to what
they might have emphasised in their teaching, the teachers were also asked how they would
wanted their students describe function and write questions to assess the student
understanding of it. Items 3 and 4 (desired) therefore referred to how the teachers thought
students should define and exemplify function. Item 5 (assessing understanding) required
that the teachers create questions that could reveal students’ understanding of function (Item
5). The teachers could give multiple responses.
Data analysis was run in two steps. First, an inductive content analysis (Elo &
Kyngäs, 2007) was implemented to identify the analogies in all 42 teachers’ responses to
define or exemplify function for Items 1 to 5. The data were scrutinised to identify all
instances of concrete-based domain comparisons (e.g., “[I would ask] Whether the
correspondence between children (domain) and mothers (range) is a function or not” T17,
Item 5). Among the total of 224 descriptions and exemplifications, this search hit a total
number of 61 (f: frequency) instances where the base domain was realistic and concrete,
representing total of 26 teachers. Two kinds of domain comparisons were identified: analogy
and anomaly. Also, a third category (Other) was evident which contained one instance of a
word problem and a few responses referencing real-life based descriptions, but no further
detail was given, such as: “I would give examples from daily life conveying the notion of
Page 12 of 26
URL: http://mc.manuscriptcentral.com/tmes Email: tmes-peerreview@journals.tandf.co.uk
International Journal of Mathematical Education in Science and Technology
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
For Peer Review Only
13
Investigating Secondary Mathematics Teachers’ Analogies to Function
function” (T13, Item 5) (see Table 4). In the responses of some teachers, there was more than
one comparison, and each was recorded: “We usually use mail and mailbox examples [the
mail-mailbox analogy]. Child and mother [the child-mother linkage analogy]” (T15, Item 4).
Next, a deductive content analysis (Elo & Kyngäs, 2007) was run to find out what
views of function those analogies illustrate. Each analogy was grouped into the views of
function presented in Table 1 that they represent. Two views of function were revealed in the
teachers’ analogies: input-output machines; and a mapping between two sets (see Table 5).
Examples of those analogies illustrating each view are provided in Table 3 and described in
detail in the following section.
Table 2: The function concept questionnaire items mapped to types of items
Item
Type
1. Imagine that you have asked your students to define the concept
of function in their own words. How do you think they will
define what a function is? Please write a few examples.
(Adapted from Breidenbach et al., 1992; Even, 1993)
Anticipated:
definition
2. Imagine that you asked your students to give some examples of
functions. What kind of examples do you think they would give?
Please write a few examples.
(Adapted from Breidenbach et al., 1992)
Anticipated:
exemplification
3. How do you think students should define the concept of function?
Please write them.
Desired:
definition
4. What kind of examples do you think students should give for
functions?
Please write them.
Desired:
exemplification
5. What questions would you ask your students in order to find out
if they understand the concept of function?
(Adapted from Cooney, 1999)
Assessing
understanding
Note. Reproduced from “Secondary mathematics teachers’ analogies to function: An
application of structure-mapping,” by V. Hatisaru, In R, Marks (Ed.), British Society for
Research into Learning Mathematics (BSRLM) Proceedings, 40(3), 05. Copyright 2021 by
BSRLM and the Author.
Page 13 of 26
URL: http://mc.manuscriptcentral.com/tmes Email: tmes-peerreview@journals.tandf.co.uk
International Journal of Mathematical Education in Science and Technology
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
For Peer Review Only
14
Investigating Secondary Mathematics Teachers’ Analogies to Function
Table 3: Examples of the analogies in the descriptions of the teachers (f=46)
View of function
Analogy
Teacher statements
Input-output
machines (f=24)
Machine (f=18)
“A function is a kind of machine. There are inputs and outputs” (T26, Item 1).
“Students would give examples such as juice machine or factory machines” (T10, Item 2).
“When we teach the concept of function to students, we give the machine example. They thus define
[function] as such” (T10, Item 1).
Factory (f=4)
“[Function is like] Producing olive oil (output) by processing olives (input) in the factory” (T26, Item 4).
“Bread factory” (T34, Item 2)
Schooling (f=1)
See T19’s response below.
Bus (f=1)
“A bus goes from Ankara to Istanbul [two metropolitan cities]” (T2, Item 2).
A mapping
between two sets
(f=20)
Child-mother linkage
(f=12)
“[A function is like] Mother-child relation (every child surely has a mother)” (T8, Item 2).
“If Set A is the set of children, Set B is the set of mothers: (i) no element will be left out in Set A as
every child has a mother; (ii) a child cannot have more than one mother [uniqueness] and also not all
mothers have children [upper bound for the range]” (T21, Item 4).
Postman (f=2)
“[Students define or exemplify functions as a] Difficult subject. Machine. Postman. Mirror.” (T1, Items
1 and 2).
Mail-mailbox (f=1)
“We usually use mail and mailbox examples. Child and mother” (T15, Item 4).
Room allocation (f=1)
“A group of students in a school camp will accommodate in a hotel and will be allocated in rooms. Some
hotel rooms may vacant [upper bound for the range], but every student must be allocated in a room. A
student can’t stay in two different rooms at the same time [uniqueness]” (T11, Item 4).
Restaurant (f=1)
“The activity in the textbook. That is, students going to a restaurant. Washing machine. A student who
becomes a professional after being educated at school” (T19, Item 2).
Dancing (f=1)
“Can be defined as a correspondence. Ali and Ayse [student names] are dancing” (T24, Item 1).
Backgammon (f=1)
“Two friends are playing backgammon” (T24, Item 2).
Cars (f=1)
“Functions are like cars departing from City A to City B. They, however, have features. First, they must
carry all people from City A to City B [uniqueness]. Second, people arriving City B should remain as
whole. That is, they may be changed/transformed but must arrive as a whole” (T16, Item 3).
Uniqueness (f=2)
Unable to be in two
places (f=1)
Fork (f=1)
“Students can think that they cannot be in somewhere else [at the same time] when they are here in the
classroom [now]” (T23, Item 4).
“There is no fork [in the domain]. i.e. each element [of the domain] is assigned one element [of the
range], none [of the domain] element is unassigned” (T34, Item 1).
Page 14 of 26
URL: http://mc.manuscriptcentral.com/tmes Email: tmes-peerreview@journals.tandf.co.uk
International Journal of Mathematical Education in Science and Technology
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
For Peer Review Only
15
Investigating Secondary Mathematics Teachers’ Analogies to Function
Results
Analogies and Anomalies in the descriptions of the Teachers
As described earlier, analogy is a comparison in which relational predicates can be mapped
from the base domain to the target domain (Gentner, 1983). This study identified several
analogies to function used by the teachers. The responses of the teachers to anticipated
student thinking (Items 1 and 2) involved more varied analogies (nine different analogies, a
total of 30 mentions), compared to the responses to desired student descriptions for function
(Items 3 and 4) (five analogies, a total of 11 mentions), and to assessing the understanding of
students (Item 5) (one analogy, 3 mentions) (see Table 4). The machine and child-mother
linkage analogies were clearly the most popular across all items.
Table 4: The concrete-based domain comparisons in the descriptions of the teachers (f=61).
Type of item
Kind of comparison:
Other (f=6)
analogy (f=46)
anomaly (f=9)
Anticipated:
definition
Machine (f=10)
Postman (f=1)
Factory (f=2)
Dancing (f=1)
Fork (f=1)
Everyday use (f=4)
Sets (f=1)
-
Anticipated:
exemplification
Machine (f=4)
Child-mother link (f=5)
Factory (f=2)
Postman (f=1)
Restaurant (f=1)
Bus (f=1)
Schooling (f=1)
Backgammon (f=1)
Everyday use (f=3)
Mirror (f=1)
-
Desired:
definition
Cars (f=1)
-
Real-life based (f=1)
Desired:
exemplification
Machine (f=4)
Child-mother link (f=4)
Mail-mail box (f=1)
Room allocation (f=1)
Unable to be in two
places (f=1)
Everyday use (f=2)
-
Assessing
understanding
Child-mother (f=3)
-
Real-life based (f=2)
Word problem (f=1)
Note. Reproduced from “Secondary mathematics teachers’ analogies to function: An
application of structure-mapping,” by V. Hatisaru, In R, Marks (Ed.), British Society for
Research into Learning Mathematics (BSRLM) Proceedings, 40(3), 05. Copyright 2021 by
BSRLM and the Author.
Page 15 of 26
URL: http://mc.manuscriptcentral.com/tmes Email: tmes-peerreview@journals.tandf.co.uk
International Journal of Mathematical Education in Science and Technology
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
For Peer Review Only
16
Investigating Secondary Mathematics Teachers’ Analogies to Function
In the Views of Function section, all provided analogies are described in detail. Each of the
examples discussed can be linked directly to the principles suggested in the structure-
mapping framework.
The more common anomalies in the responses of the teachers (f=9) were those
describing function as what something is used for, or the actions and activities assigned to or
required or expected of a person (Wolfram Alpha LLC., 2020). Illustrative examples were:
“[Students think] Function. The function of technological machines” (T19, Item 1) and “The
function of a telephone; the function of a teacher in the classroom” (T41, Item 2), where
there need be no analogical matches, but only a literal similarity. Other examples included
colloquial statements similar to those given by Tabach and Nachlieli (2015), (see the
following paragraph). It is important to note that seven of those occurrences were the
teachers’ anticipation of student (unsatisfactory) descriptions and exemplifications to
function (Author et al., 2013).
As elaborated in Author (2021), the teachers’ anticipation of student (mis)thinking
(Items 1 and 2) also included the mirror (see Table 3, T1) and sets (unsatisfactory)
comparisons (“Students think that function is topic related to sets. For instance, the set of
blond students in the class” T23, Item 1). A mirror or sets convey few relational
commonalities with functions, and as there might be only common object attributes between
the two (e.g., functions have domain and range sets), such a comparison can be considered
only a surface similarity. Two teachers, however, wrote that students should exemplify
function: “Related to real life, devices they use, computer, phone” (T1, Item 4) and “One can
lose their life functions. A tool may have various functions. f(x), x” (T25). It seems these two
teachers would be satisfied by students responding to function as if it were a word, through
its everyday meaning. This potential unsatisfactory thinking of the teachers, though
important, is difficult to discern from the current data in the absence of interviews. Thus, I
have made no analytical claims about it.
Views of Function
The comparisons made by the teachers corresponded mostly to either the view of a function
as input-output machines or a mapping between two sets (see Table 5), and these views (both
refer to the correspondence approach) dominated their analogical reasoning. Some illustrative
examples representing both views are presented in Table 3.
Page 16 of 26
URL: http://mc.manuscriptcentral.com/tmes Email: tmes-peerreview@journals.tandf.co.uk
International Journal of Mathematical Education in Science and Technology
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
For Peer Review Only
17
Investigating Secondary Mathematics Teachers’ Analogies to Function
Function as an input-output machine
In the analogies representing function as input-output machines, intended inferences mainly
concerned the relational structure such as a function receives some input and gives an output,
just as a factory or machine. Another relation matching the base with the target was a
function does not have to work only with numbers, like a machine or factory (arbitrariness).
That was clearly indicated in T26’s comparison between a function and an olive oil factory in
Table 3. This group also involved the schooling and bus analogies where the teachers made
comparisons between function and formal schooling (also presented in the textbook) or bus.
The intended inference seemed to be a function receives inputs and gives an appropriate
output, like schools or buses receive students or passengers and educate them or carry them
from one point to another. Among these examples, while in ten analogies the domain and
range have concrete, realistic values (e.g., input: fruits, olives, and wheat; output: juice, olive
oil, and flour), in two, the domain and range have sets of numbers.
Table 5: The view of function represented in the analogies (f=44).
View of
Anticipated:
Desired:
Assessing
function
definition
exemplification
definition
exemplification
understanding
Input-
output
machines
Machine (f=10)
Factory (f=2)
Machine (f=4)
Factory (f=2)
Schooling (f=1)
Bus (f=1)
-
Machine (f=4)
-
A mapping
between
two sets
Postman (f=1)
Dancing (f=1)
Child-mother
linkage (f=5)
Postman (f=1)
Restaurant (f=1)
Backgammon
(f=1)
Cars
(f=1)
Child-mother
linkage (f=4)
Mail-mail box
(f=1)
Room
allocation (f=1)
Child-mother
linkage (f=3)
Function as a mapping between two sets
Within this group, the child-mother linkage analogy was the most popular. In this
comparison, similarly, relations had priority over attributes of the objects. For example, a
function maps each element of one set to exactly one element of a second set (uniqueness),
just as the linkage between a mother and her child, but not function is functionality of things
or individuals like function of a mother for her child. In this analogy, the teachers had a
function f defined by domain (the set of mothers), range (the set of their children), and
definition (f(mother)=her child). An example could be f(Mother A)=Child A. As suggested
by Davidenko (1999), if we ask students: Why you did not mention Mother C? they may say:
Page 17 of 26
URL: http://mc.manuscriptcentral.com/tmes Email: tmes-peerreview@journals.tandf.co.uk
International Journal of Mathematical Education in Science and Technology
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
For Peer Review Only
18
Investigating Secondary Mathematics Teachers’ Analogies to Function
Mother C does not have a child! Here, they suggest an upper bound for the range of this
function. How about f(Mother A)=Child A and f(Mother A)=Child B? Students may say:
Mother A has two children: Child A and Child B! That is, the linkage allows many-to-one
mapping (Mother A can have more than one child) just as functions do. In responding to Item
4, T37 explicitly made these communalities in their descriptions: “Let us say the domain set
is children, the range set is mothers. Every child has a mother. A child cannot have more
than one mother [uniqueness]. But not all women have to have a child [upper bound for the
range]”, as well as in T21’s descriptions in Table 3.
As reported in Author (2021), other analogies revealing the uniqueness and upper
bound for the range similarities were the room allocation and restaurant analogies (see
responses of T11 and T19 respectively in Table 3). Like the child-mother linkage analogy,
both analogies are relational comparisons between base (hotels and restaurants) and target
(function) domains, involving relational similarities. The room allocation analogy, made by
T11, maps the domain of a function to a group of students camping, the range to rooms in a
hotel, the uniqueness feature of functions to every student must be allocated to a room, but a
student can’t stay in two different rooms, and the upper bound for the range feature to some
hotel rooms may be vacant. The restaurant analogy, established by T19, which was also
given in the textbook (MoNE, 2012), maps the domain to a group of students in a restaurant,
the range to the list of dishes that they could order, and the uniqueness to the condition that
each student could order exactly one dish.
While the condition of uniqueness is implicit in the input-output machines view of
functions, it is relatively explicit in the mapping between two sets view. In the latter group,
the uniqueness condition was a central element of the treatment of functions. The fork
analogy of T23 and T34 (see Table 3), in fact, conveys a great sense of the uniqueness
condition.
Because of their wide range of types, analogies can be challenging (Gentner et al.,
2001). For instance, although the intended inferences in the dancing and backgammon
analogies in this group seemed to be, a function is a mapping, as when two people dance or
play backgammon, they are a little unclear. In the cars analogy (T16, Table 3), the relational
structure applied to the base domain (cars) can be applied to the target domain (function) but
does not overlap strongly, and the analogy sounds unclear and invalid. As they were the
teachers’ anticipation of student thinking, through these analogies, the teachers might address
incomplete understanding of students.
Page 18 of 26
URL: http://mc.manuscriptcentral.com/tmes Email: tmes-peerreview@journals.tandf.co.uk
International Journal of Mathematical Education in Science and Technology
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
For Peer Review Only
19
Investigating Secondary Mathematics Teachers’ Analogies to Function
As given in Table 1, mappings and input-output models illustrate the correspondence
approach to functions, while referencing the manner in which a change in one variable
connects to a change in another, and how the variables change together illustrates the
covariation perspective (Ayalon et al., 2017). The covariation view was articulated only by
one teacher in the context of a word problem: “Every year a tree grows 5 cm more than twice
its age. If its height was 10 cm when it was planted, what would its height be after three
years?” (T25, Item 5).
Discussion
This study investigates the analogies used by a sample of secondary mathematics teachers
when they describe the concept of function: how students would (anticipated) and should
(desired) define and exemplify function; and what questions the teachers would ask to assess
students’ understanding of function (assessing understanding). The analogies in the
descriptions of the participant teachers are identified and ways in which these are structurally
mapped to function are examined by applying Gentner’s structure-mapping theory of
analogy. The study has revealed that teachers’ repertoire is rich in analogy and includes
interesting varieties of a popular child-mother linkage analogy for reasoning about functions.
As shown by examples such as a function is like mother-child relation, every child
surely has a mother or functions are kind of machines, there are inputs and outputs, analogy
is a compelling tool for capturing the different ways in which teachers ascribe the mapping to
functions. The presentation of information as a (biological) linkage between a child and
mother, or allocation of a group of students to hotel rooms, has been consistent with Sand’s
(1999) account of the mail carrier analogy in teaching functions. This view of function is
useful to show students that the domain and range set of functions do not necessarily have to
be sets of numbers, to assist students in building a concrete understanding of functions, and to
promote the internalisation of the uniqueness condition of functions (Sand, 1999).
Nevertheless, the covariation and dependency characteristics of functions are not well
described by these analogies. Although constant, one-to-one, and onto functions can be
illustrated in this way, function types such as continuous, increasing, or quadratic functions
contain ordered numerical domain and range requirements, and so analogies such as the mail-
carrier (or, in this study, the child-mother linkage) do not describe them well (Sand, 1999).
To that end, although they are valid and clear, they are limited in scope. It can be argued that
Page 19 of 26
URL: http://mc.manuscriptcentral.com/tmes Email: tmes-peerreview@journals.tandf.co.uk
International Journal of Mathematical Education in Science and Technology
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
For Peer Review Only
20
Investigating Secondary Mathematics Teachers’ Analogies to Function
the participant teachers think in that way because this sort of comparison is available and
appears to serve as a communicative tool for the teachers. Also, sometimes analogies which
cover the broad range of targets whilst maintaining firm definitions of objects and predicates
can be hard to find (Gentner, 1982). However, it might be useful for teachers to be alerted to
the conceptions of function they emphasis in their teaching, and to ask themselves why they
use or generate the analogies that they do. Teachers might be invited to form alternative
analogies and to reflect upon the impact that the alternatives may have upon the way they
think and teach about functions.
The analogies present in the teachers’ descriptions provide detailed information about
how teachers understand function. Also, it is likely that the teachers use these analogies in
their teaching about functions (see responses of T10 and T15 in Table 3), like participants in
some other studies (see Espinoza-Vásquez et al., 2017; Author et al., 2017; Ubuz et al.,
2009). As in many high school mathematics curricula and mathematics textbooks (Cooney et
al., 2010), uniqueness was the main element in the treatment of function (see also Author,
2020). The correspondence approach to function (mappings and input-output machines) was
dominant in the teachers’ responses while the covariation approach was entirely absent. The
analogies used represent a more widespread curriculum influence of referring to functions as
taking inputs to outputs (operations or manipulations) or to a mapping from one set to
another. I therefore conjecture that the teachers’ conceptions revealed in their analogies might
have been influenced by the curriculum and associated textbook (Ayalon et al., 2017; Davis,
2009). It might also be that the choice of analogical comparison is a matter of its aptness.
Following that, the issue becomes the depth of CCK about functions held by the teachers or
their knowledge of how to communicate their knowledge for the purpose of teaching about
them (SCK). This shows that an analogy represents something of how teachers understand
functions (CCK) or think that the analogy (e.g., the mother-child linkage) would be the one
that can help students to understand (SCK). This study unfortunately cannot uncover why the
participant teachers predominantly selected analogies that represent the correspondence
conception of functions from several other available ones representing the covariation
conception, and this is an important issue for future research.
Page 20 of 26
URL: http://mc.manuscriptcentral.com/tmes Email: tmes-peerreview@journals.tandf.co.uk
International Journal of Mathematical Education in Science and Technology
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
For Peer Review Only
21
Investigating Secondary Mathematics Teachers’ Analogies to Function
Limitations and Future Directions
While the findings of this study provide useful information about how teachers may view,
understand, and teach the concept of function in the classroom, there is at least two
limitations that should be considered. First, I only studied a sample of Turkish voluntary
secondary mathematics teachers’ responses to five open-ended questionnaire items, and I did
not observe their classroom teaching. Second, it is uncertain whether the Turkish teachers’
analogies may appear in the descriptions of other teachers in other cultural settings. I am fully
aware that differences exist in teachers’ mathematical knowledge or beliefs related to
mathematics or its learning and teaching in different cultural settings (e.g., An et al., 2004). I
would therefore be cautious about generalisations of the findings in this study to other
countries. An extension of this study might be to investigate how teachers in Turkey and
other countries view the concept of function and what analogies they construct to explain this
concept.
Whilst I would be cautious about generalisations of the findings, the conceptual and
methodological approaches used in this study contains important implications and directions
for future research. First, the study contributes our current understanding of the notion of
analogy in the functions content domain as it deals in detail with analogies in the descriptions
of teachers about the concept of function, a matter not typically addressed (see exceptions
Espinoza-Vásquez et al., 2017; Ubuz et al., 2009). Second and most significantly, the study
develops and presents a conceptualisation that can be used to analyse teachers’ MKT about
functions, and beyond. The teachers’ analogies identified in this study, as possibly
representation of their understanding, contain key characteristics of the concept of function.
Consistent with Author’s (2020) finding which revealed that uniqueness was the main
element for participant teachers in the treatment of function, many of the teachers viewed
function as a special type of relation, and, for T23 in particular, this feature seems to be
caught up with T23’s view of understanding and not understanding the concept of function.
Many of the teachers’ analogical arguments included features of functions such as the
uniqueness and upper bound for the range, and they can be therefore seen to be related to the
concept of function. What is not known and requires task-based interviews or classroom
observations to discover is the ways that analogies are used in the classroom, elaborations of
the analogies, and how students are engaged in complex relational reasonings (Richland et
al., 2007).
Page 21 of 26
URL: http://mc.manuscriptcentral.com/tmes Email: tmes-peerreview@journals.tandf.co.uk
International Journal of Mathematical Education in Science and Technology
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
For Peer Review Only
22
Investigating Secondary Mathematics Teachers’ Analogies to Function
Acknowledgements
An early version of this paper was presented at the British Society for Research into Learning
Mathematics (BSRLM) November 2021 Online Day Conference. I am grateful for the
teachers participated in this study, and the Ministry of National Education which approved
the study implementation. Any opinions presented herein are yet those of mine. I
acknowledge the feedback and editorial assistance from the International Journal of
Mathematical Education in Science & Technology, and reviewers, and Emily Morgan for
proofreading the paper. All these efforts contributed to the paper’s improvement, and of
course, any errors remain solely the responsibility of myself.
Disclosure statement
No potential conflict of interest was reported by the authors.
References
Author et al., (2013). Kastamonu Education Journal.
Author et al., (2017). International Journal of Science and Mathematics Education.
Author (2020). International Journal for Mathematics Teaching and Learning.
Author (2021). BSRLM conference paper.
An, S., Kulm, G., & Wu, Z. (2004). The pedagogical content knowledge of middle school,
mathematics teachers in China and the U.S. Journal of Mathematics Teacher Education,
7, 145-172.
Aubusson, P. J., Harrison, A. G., & Ritchie, S. M. (2006). Metaphor and analogy: Serious
thought in science education. In P. J. Aubusson, A. G. Harrison, & S. M. Ritchie (Eds.).
Metaphor and analogy in science education, (pp. 1-9). Springer: Dordrecht, The
Netherlands.
Ayalon, M., Watson, A., & Lerman, S. (2017). Students’ conceptualisations of function
revealed through definitions and examples. Research in Mathematics Education, 19(1),
1-19. doi:10.1080/14794802.2016.1249397
Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: what
makes it special? Journal of Teacher Education, 59(5), 389-407.
Bowdle, B. F., & Gentner, D. (2005). The career of metaphor. Psychological Review, 112(1),
193-216. doi: 10.1037/0033-295X.112.1.193
Breidenbach, D., Dubinsky, E., Hawks, J., & Nichols, D. (1992). Development of the process
conception of function. Educational Studies in Mathematics, 23(3), 247-285.
Brown, J. S., Collins, A., & Duguid P. (1989). Situated cognition and the culture of learning.
Educational Researcher, 18(1), 32-34.
Clement, L. (2001). What do students really know about functions? The Mathematics
Teacher, 94(9), 745-748.
Cooney, T. (1999). Conceptualizing teachers’ ways of knowing. Educational Studies in
Mathematics, 38, 163-187.
Page 22 of 26
URL: http://mc.manuscriptcentral.com/tmes Email: tmes-peerreview@journals.tandf.co.uk
International Journal of Mathematical Education in Science and Technology
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
For Peer Review Only
23
Investigating Secondary Mathematics Teachers’ Analogies to Function
Cooney, T. J., & Wilson, M. R. (1993). Teachers' thinking about functions: Historical and
research perspectives. In T. A. Romberg, E. Fennema, & T. P. Carpenter (Eds.),
Integrating research on the graphical representation of functions (pp. 131-158). Hillside,
NJ: Lawrence Erlbaum.
Cooney, T. J., Beckmann, S., Lloyd, G. M., Wilson, P. S., & Zbiek, R. M. (2010).
Developing essential understandings of functions for teaching mathematics in grades 9-
12. Reston, VA: National Council of Teachers of Mathematics (NCTM).
Davidenko, S. (1999). Building the concept of function from students’ everyday activities. In
B. Moses (Ed.), Algebraic thinking grades K-12. Readings from NCTM’s school-based
journals and other publications (pp. 206-212). NCTM, Reston, Virginia.
Davis, J. (2009). Understanding the influence of two mathematics textbooks on prospective
secondary teachers’ knowledge. Journal of Mathematics Teacher Education, 12(5), 365-
389.
Duit, R. (1991). On the role of analogies and metaphors in learning science. Science
Education, 75, 649–672.
Dubinsky, E., & Wilson, R. T. (2013). High school students’ understanding of the function
concept. The Journal of Mathematical Behavior, 32(1), 83-101.
Elo, S., & Kyngäs (2008). The qualitative content analysis process. Journal of Advanced
Nursing, 62(1), 107-115. doi:10.1111/j.1365-2648.2007.04569.x
English, L. (1997). Analogies, metaphors, and images: vehicles for mathematical reasoning.
In Mathematical Reasoning: Analogies, Metaphors, and Images. Lawrence Erlbaum
Associates, New Jersey, USA, pp. 3-18.
English, L. D. (2015). STEM: challenges and opportunities for mathematics education. In K.
Beswick, T. Muir, & J. Wells (Eds.), Proceedings of the 39th Conference of the
International Group for the Psychology of Mathematics Education (Vol. 1, pp. 3-18).
Hobart, Australia: PME.
Espinoza-Vásquez, G., Zakaryan, D., Yañez, J. C. (2017). Use of analogies in teaching the
concept of function: relation between knowledge of topics and knowledge of
mathematics teaching. CERME Thematic Working Group 20. Dublin, Ireland. hal-
01949154
Even, R. (1990). Subject matter knowledge for teaching and the case of functions.
Educational Studies in Mathematics, 21(6), 521-554.
Even, R. (1993). Subject-matter knowledge and pedagogical content knowledge: prospective
secondary teachers and the function concept. Journal for Research in Mathematics
Education, 24(2), 94-116.
Even, R., & Tirosh, D. (1995). Subject-matter knowledge and the knowledge about students
as sources of teacher presentations of the subject-matter. Educational Studies in
Mathematics, 29(1), 1-20.
Gentner, D. (1982). Are scientific analogies metaphors? In D. Miall (Ed.), Metaphor:
Problems and Perspectives. Brighton, Sussex: Harvester Press, Ltd.
Gentner, D. (1983). Structure-mapping: a theoretical framework for analogy. Cognitive
Science, 7(2), 155-170. doi:10.1016/S0364-0213(83)80009-3
Gentner, D. (1989). The mechanisms of analogical learning. In S. Vosniadou & A. Ortony
(Eds.), Similarity and analogical reasoning (pp. 199-241). London: Cambrige University
Press.
Gentner, D., & Markman, A. B. (1997). Structure mapping in analogy and similarity.
American Psychologist, 52(1), 45-56. doi:10.1037/0003-066X.52.1.45
Page 23 of 26
URL: http://mc.manuscriptcentral.com/tmes Email: tmes-peerreview@journals.tandf.co.uk
International Journal of Mathematical Education in Science and Technology
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
For Peer Review Only
24
Investigating Secondary Mathematics Teachers’ Analogies to Function
Gentner, D., & Maravilla, F. (2018). Analogical reasoning. In L. J. Ball, & V. A. Thompson
(Eds.), International Handbook of Thinking & Reasoning (pp. 186-203). NY, NY:
Psychology Press.
Gentner, D., Bowdle, B., Wolff, P., & Boronat, C. (2001). Metaphor is like analogy. In
Gentner, D., Holyoak, K. J., & Kokinov, B. N. (Eds.), The analogical mind: Perspectives
from cognitive science (pp. 199-253). Cambridge MA, MIT Press.
Glynn, S. M. (2008). Making science concepts meaningful to students: teaching with
analogies. In S. Mikelskis-Seifert, U. Ringelband, and M. Bruckmann, (Eds.), Four
decades of research in science education: From curriculum development to quality
improvement (pp. 113-125). Waxmann, Mnster, Germany.
Hill, H. C., Blunk, M. L., Charalambous, C. Y., Lewis, J. M., Phelps, G. C., Sleep, L., &
Ball, D. L. (2008). Mathematical knowledge for teaching and the mathematical quality of
instruction: an exploratory study. Cognition and Instruction, 26(4), 430-511. doi:
10.1080/07370000802177235
Hitt, F. (1998). Difficulties in the articulation of different representations linked to the
function concept. Journal of Mathematical Behavior, 17(1), 123-134.
Lakoff, G., & Núñez, R. (2000). Where mathematics comes from: how the embodied mind
brings mathematics into being. Basic Books.
James, M. C., & Scharmann, L. C. (2007). Using analogies to improve the teaching
performance of preservice teachers. Journal of Research in Science Teaching, 44(4),
565-585. doi:10.1002/tea.20167
Meiers, M. & Trevitt, J. (2010). Language in the mathematics classroom. The Digest, NSWIT,
Edition 2010/2. Retrieved from http://www.nswteachers.nsw.edu.au
McCulloch, A. W., Lovett, J. N., Meagher, M. S., & Sherman, M. F. (2020). Challenging
preservice secondary mathematics teachers’ conceptions of function. Mathematics
Education Research Journal. doi:10.1007/s13394-020-00347-6
Ministry of National Education. (2012). Secondary mathematics: textbook for grade 9.
Istanbul, Turkey: Devlet Kitapları.
Ministry of National Education. (2019). Secondary mathematics: textbook for grade 10.
Istanbul, Turkey: Devlet Kitapları.
Nyikahadzoyi, M. R. (2015). Teachers’ knowledge of the concept of a function: a theoretical
framework. International Journal of Science and Mathematics Education, 13(2), 261-
283.
Ribeiro, A. J., & da Ponte, J. P. (2019). Professional learning opportunities in a practice-
based teacher education programme about the concept of function. Acta Scientiae,
21(2), 49-74. doi:10.17648/acta.scientiae.v21iss2id5002
Richland, L. E., Zur, O., & Holyoak, K. J. (2007). Cognitive supports for analogies in the
mathematics classroom. Science, 316, 1128-1129.
Richland, L. E., & McDonough, I. M. (2010). Learning by analogy: discriminating between
potential analogs. Contemporary Educational Psychology, 35, 28-43.
doi:10.1016/j.cedpsych.2009.09.001
Rodríguez-Nieto, C. A., Rodríguez-Vásquez, F. M., & Moll, V. F. (2020): A new view about
connections: the mathematical connections established by a teacher when teaching the
derivative. International Journal of Mathematical Education in Science and
Technology. doi:10.1080/0020739X.2020.1799254
Roux, A., Nieuwoudt, H., & Nieuwoudt, S. (2015). Conceptual learning of functions in a
technologically enhanced environment. African Journal of Research in Mathematics,
Science and Technology Education, 19(3), 289-305.
Page 24 of 26
URL: http://mc.manuscriptcentral.com/tmes Email: tmes-peerreview@journals.tandf.co.uk
International Journal of Mathematical Education in Science and Technology
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
For Peer Review Only
25
Investigating Secondary Mathematics Teachers’ Analogies to Function
Rowland, T. (2013). The knowledge quartet: the genesis and application of a framework for
analysing mathematics teaching and deepening teachers’ mathematics knowledge.
Sisyphus Journal of Education, 1(3), 15-43.
Sand, M. (1999). A function is a mail carrier. In B. Moses (Ed.), Algebraic thinking grades
K-12. Readings from NCTM’s school-based journals and other publications (pp. 213-
219). NCTM, Reston, Virginia.
Shulman, L. (1986). Those who understand: knowledge growth in teaching. American
Educational Research Association, 15(2), 4-14.
Steele, M. D., Hillen, A. F., & Smith, M. S. (2013). Developing mathematical knowledge for
teaching in a methods course: the case of function. Journal of Mathematics and Teacher
Education, 16(6), 451-482.
Stein, M. K., Baxter, J. A., & Leinhardt, G. (1990). Subject-matter knowledge and
elementary instruction: A case from functions and graphing. American Educational
Research Journal, 27(4), 639-663.
Stover, C., & Weisstein, E. W. (2017). Function. In MathWorld—A Wolfram Web Resource.
Retrieved from http://mathworld.wolfram.com/Function.html
Son, J., & Hu, Q (2015). The initial treatment of the concept of function in the selected
secondary school mathematics textbooks in the US and China. International Journal of
Mathematical Education in Science and Technology.
doi:10.1080/0020739X.2015.1088084
Tabach, M., & Nachlieli, T. (2015). Classroom engagement towards using definitions for
developing mathematical objects: the case of function. Educational Studies in
Mathematics, 90, 163-187. doi:10.1007/s10649-015-9624-0
Turkish Board of Education. (2011). Secondary school mathematics syllabus (Grades 9, 10,
11, and 12). Ankara, Turkey: Author.
Thompson, P. W., & Carlson, M. P. (2017). Variation, covariation, and functions:
Foundational ways of thinking mathematically. In J. Cai (Ed.), Compendium for research
in mathematics education (pp. 421-456). Reston, VA: National Council of Teachers of
Mathematics. Retrieved from http://pat-
thompson.net/PDFversions/2016ThompsonCarlsonCovariation.pdf
Thorpe, J. A. (1999). Algebra: What should we teach and how should we teach it. In B.
Moses (Ed.), Algebraic thinking grades K-12. Readings from NCTM’s school-based
journals and other publications (pp. 22-30). NCTM, Reston, Virginia.
Oehrtman, M. C., Carlson, M. P., & Thompson, P. W. (2008). Foundational reasoning
abilities that promote coherence in students' understandings of function. In M. P. Carlson
& C. Rasmussen (Eds.), Making the connection: Research and practice in
undergraduate mathematics (pp. 27-42). Washington, DC: Mathematical Association of
America.
Ubuz, B., Eryılmaz, A., Aydın, U., & Bayazıt, I. (2009). Pre-service teacher generated
analogies for function concepts. In V. Durand-Guerrier, S. Soury Lavergne, & F.
Arzarello (Eds.), Proceedings of the Sixth Congress of the European Society for
Research in Mathematics Education (pp.1871-1879). Lyon: France.
Unver, E. (2009). Analysis of analogy use on function concept in the ninth-grade
mathematics textbook and classrooms (Master dissertation). Retrieved from
http://etd.lib.metu.edu.tr/upload/3/12611356/index.pdf
Watson, A., & Harel, G. (2013). The role of teachers’ knowledge of functions in their
teaching: a conceptual approach with illustrations from two cases. Canadian Journal of
Science, Mathematics, and Technology Education, 13(2), 154-168.Wilkie, J. W. (2014).
Upper primary school teachers’ mathematical knowledge for teaching functional
Page 25 of 26
URL: http://mc.manuscriptcentral.com/tmes Email: tmes-peerreview@journals.tandf.co.uk
International Journal of Mathematical Education in Science and Technology
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
For Peer Review Only
26
Investigating Secondary Mathematics Teachers’ Analogies to Function
thinking in algebra. Journal of Mathematics Teacher Education, 17(5), 397-428. doi:
0.1007/s10857-013-9251-6
Wolfram Alpha LLC. (2020). Wolfram|Alpha. Accessed from
https://www.wolframalpha.com/input/?i=function (access November, 12, 2020).
Wilkie, J. W. (2014). Upper primary school teachers’ mathematical knowledge for teaching
functional thinking in algebra. Journal of Mathematics Teacher Education, 17(5), 397-
428. doi: 0.1007/s10857-013-9251-6
Willoughby, S. S. (1997). Activities to help in learning about functions. Mathematics
Teaching in the Middle School, 2(4), 215-219.
Zazkis, R., & Marmur, O. (2018). Scripting tasks as a springboard for extending prospective
teachers’ example spaces: a case of generating functions. Canadian Journal of Science,
Mathematics and Technology Education, 18(4), 291–312. doi:10.1007/s42330-018-
0019-y
Page 26 of 26
URL: http://mc.manuscriptcentral.com/tmes Email: tmes-peerreview@journals.tandf.co.uk
International Journal of Mathematical Education in Science and Technology
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
