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Received : 02.11.2022
Published : 10.03.2022
Available Online : 30.04.2023
Interpretations of Pre-service Elementary Mathematics Teachers on
the Functions of Non-Textual Elements: Case Study on Algebra
Learning Area
Murat GENÇa Mustafa AKINCIb
a:
0000-0003-4525-7507
Zonguldak Bülent Ecevit University,
Turkiye
muratgenc@beun.edu.tr
b:
0000-0003-2096-7617
Zonguldak Bülent Ecevit University,
Turkiye
mustafa.akinci@beun.edu.tr
Ethics Committee Approval: Ethics committee permission for this study was obtained from Zonguldak
Bülent Ecevit University Human Research Ethics Committee with the decision dated 05.10.2022 and
numbered 221989/326.
Suggested Citation: Genç, M., & Akıncı, M. (2023). Interpretations of Pre-service Elementary
Mathematics Teachers on the Functions of Non-Textual Elements: Case Study on Algebra Learning
Area. Sakarya University Journal of Education, 13(1), 84-102. doi: https://doi.org/10.19126/suje.1200724
Abstract
The study aimed to investigate how pre-service elementary mathematics teachers perceive the intended use
of non-textual elements in an algebra content area of an eighth-grade mathematics textbook. Non-textual
elements in this qualitative exploratory case study refer to visual representations consisting of components
that are not only verbal, numerical, or symbolic representations. Data were collected from thirty-one
undergraduate students through a task-based written questionnaire including seven non-textual elements
on the algebra learning domain. Data analysis was conducted using a content analysis approach to generate
themes and uncover previously unspecified patterns. The results showed that pre-service teachers’
interpretations of non-textual elements could be categorized into ten themes: (i) attractiveness, (ii)
organizing, (iii) embodiment, (iv) informativeness, (v) reasoning, (vi) conciseness, (vii) essentiality, (viii)
decorativeness, (ix) contextuality, and (x) connectivity. Pre-service teachers were found to have diverse but
sometimes overlapping interpretations of the functions of each non-textual element. However, the
functional diversity of non-textual elements may have differentiated their interpretations, as visual literacy
skills and strategies are required to interpret the intended use of non-textual elements. Therefore, in order
for pre-service mathematics teachers to better understand the functions of non-textual elements, various
teaching approaches should be developed to support pre-service teachers’ visual literacy, and these
approaches to visual literacy should be incorporated into teacher education and professional development.
Keywords
Non-textual elements, mathematics textbook, algebra learning area, pre-service elementary mathematics
teachers.
Volume: 13 Issue: 1 – Sakarya University Journal of Education ● 85
INTRODUCTION
Attracting students’ visual attention is essential for teaching, communicating key concepts, and
engaging students emotionally with course content (Araya, Farsani, & Hernández, 2016). Since visual
representations can capture students’ attention and have a positive impact on learning outcomes
(Levin & Mayer 1993; Pettersson, 1990; Woodward, 1993), it can be seen that more images,
illustrations, and diagrams are included in textbooks than in the past (Bazerman, 2006; Boling, Eccarius,
Smith & Frick 2004). On the other hand, it has also been reported that visual images, particularly in
science and reading books, can sometimes make it difficult for students to understand the content and
even lead to confusion and misinterpretation of the text (Watkins, Miller & Brubaker, 2004). However,
due to the abstract nature of mathematics, it is argued that representing or illustrating mathematical
concepts or situations in different ways helps students develop their abstract thinking skills and
contributes to mathematical understanding (National Council of Teachers of Mathematics [NCTM],
2014).
Visualization is not a new method for mathematicians, and they have long been aware of this method
and have made great efforts to take advantage of it (Borwein & Jörgenson, 2001). Visualization makes
it easy to understand how mathematical ideas are structured and how they relate to each other
(Farmaki & Paschos, 2007). Since representations are directly related to both mathematical content
and the learning process, they are considered to be effective in the formation of an individual’s concept
image, mathematical communication, and reasoning (Hershkowitz, Arcavi & Bruckheimer, 2001; Tall
& Vinner 1981). Therefore, in addition to textual elements such as written texts, mathematical signs,
and symbols, non-textual representations such as figures, diagrams, and various pictures are believed
to play an important role in teaching mathematics (Arcavi, 2003; Brenner et al., 1997; Herman, 2007;
Pape & Tchoshanov, 2001; Stylianou & Silver, 2004).
Algebra is a critical content area of focus in mathematics teaching as it provides important
opportunities for the development of mathematical thinking, reasoning, and problem-solving skills
(Van de Walle, Karp & Bay-Williams, 2012). NCTM’s vision for school mathematics recognizes the
importance of algebra and highlights why all students should learn algebra. First of all, algebra uses
abstract structures and the principles of these structures in solving problems expressed with symbols.
Besides, the ideas contained in the algebra standard are an important component of the school
mathematics curriculum and help unify its content areas. For example, most of the symbolic and
structural emphasis in algebra is built on students’ knowledge of numbers. Algebra is also closely
related to geometry and data analysis. In addition, algebraic competence is important in individuals’
further education and later in their working lives. Moreover, algebra represents patterns in our daily
lives and generalizes arithmetic. In other words, it is the language of generalization used to create a
systematic representation of patterns and relationships between numbers and objects, to analyze
change, and to model real-world events (NCTM, 2000, 2018).
The concepts of algebra can be studied and communicated through representations, enabling students
to interpret relationships among quantities and make sense of symbols (Kieran, 2004). Being successful
in algebra depends on algebraic thinking, defined as the individual’s ability to generalize about
mathematical operations and relationships/patterns, to make assumptions from these generalizations,
and to discuss and express them (Kaput, 1999). Since algebraic thinking is expressed as the use of
mathematical symbols and tools to represent verbally expressed mathematical knowledge with
figures, tables, graphs, and equations by selecting the necessary information from the given problem
86 ● Genç, M., & Akıncı, M.
situation (Herbert & Brown, 1997), there needs to be the effective use of multiple representations and
relating these representations with each other in teaching algebra. Indeed, the use of representations
in mathematics teaching is widely considered necessary, because abstract mathematical ideas,
concepts, or relationships can only be accessed through representations and their effective use in
teaching (Duval, 2006). On the other hand, a variety of representations appear to be readily available
and widely used in curriculum materials, but research encourages educators to carefully examine both
their benefits and limitations to support students’ learning (Kamii, Lewis & Kirkland, 2001), rather than
having a high expectation that the anticipated functions of these representations will somehow occur
spontaneously (Ball, 1992). Mathematics curriculum materials include not only textual elements such
as standard text, mathematical signs and symbols, but also various non-textual elements such as
figures, tables, graphs, diagrams, pictures, images, and illustrations (Filloy, Rojano & Puig, 2008).
Teachers are also expected to constantly interact with curriculum materials to assist and guide their
teaching, including textbooks, teacher guides, student worksheets, and other types of resources (Stein,
Remillard & Smith, 2007). However, if teachers interpret and apply representations inappropriately,
incorrect messages may be transmitted and then basic mathematical concepts may be distorted, which
can further confuse students (Bosse, Lynch-Davis, Adu-Gyamfi & Chandler, 2016). For this reason,
teachers need to use various non-textual elements effectively for students to learn mathematics
meaningfully. Despite the importance of non-textual elements in teaching and learning mathematics
and the potential for misuse, their features and roles in mathematics curriculum materials are still
elusive for many teachers, especially for prospective teachers (Lee & Ligocki, 2020). Therefore, this
paper aims to explore pre-service teachers’ interpretation of the functions of non-textual elements in
a mathematics textbook. More specifically, the study attempts to investigate how pre-service
elementary mathematics teachers perceive the characteristics and roles of non-textual elements
selected from an algebra content area of an eighth-grade mathematics textbook.
Literature Review and Conceptual Background
In this study, we are particularly concerned with the intended use and overall quality of non-textual
elements referring to visual representations consisting of components that are not purely verbal,
numerical, or mathematical symbolic representations (Kim, 2009). For example, the equation
2 2 2
a b c+=
used in the Pythagorean Theorem is not a non-textual element, as it consists only of
symbolic representations. However, if it is illustrated with a right triangle picture in which the
necessary symbols and signs are used, the picture is a non-textual element because it is not a purely
symbolic representation despite some symbols and signs it contains. Kim (2012) emphasizes that
research on visual representations in mathematics textbooks usually focuses on mathematical
representations such as formulae, numbers, tables, graphs, charts, diagrams, symbolic equations, and
the like. On the other hand, pictorial representations such as pictures, drawings, photos, and
illustrations are often considered decorations or a part of the visual design of a textbook. To provide a
more systematic understanding of non-textual elements in mathematics textbooks, Kim (2009, 2012)
has developed a conceptual framework, each conceptual component of which is based on various
studies on mathematics education, semiotics, metaphor theory, visual rhetoric, and information
design. In this conceptual framework, the important aspects that constitute the quality of non-textual
elements in mathematics textbooks are identified as accuracy, connectivity, contextuality,
conciseness, and aesthetics. Accuracy refers to the mathematical clarity and precision of non-textual
elements according to the definition of a mathematical concept. Connectivity signifies how closely the
non-textual elements are related to the mathematical content contained in the texts. Contextuality
Volume: 13 Issue: 1 – Sakarya University Journal of Education ● 87
indicates the presentation of mathematical expressions in a realistic context. Conciseness means
mathematical simplicity in a non-textual element. Aesthetics implies the visual appeal of non-textual
elements to facilitate and motivate learning.
In addition, seven functions of explicative illustrations distinguished by Duchastel and Waller (1979)
are stated as descriptive, expressive, constructional, functional, logico-mathematical, algorithmic, and
data-display. Descriptive denotes the function of a visual element to provide information about what
a described object actually looks like. Expressive refers to a function that aims to make an impact on
the learner beyond simple explanation. Constructional indicates the function of describing how the
various components of an object fit together to form the whole. Functional represents the function
that aims the learner to understand how a process or system works. Logico-mathematical is a function
of displaying diagrams, figures, drawings, and graphs used to explain mathematical relationships.
Algorithmic illustrates the function that provides a holistic picture of the various possibilities for an
action plan. Data-display refers to the function that provides a quick visual comparison and easy access
to data.
Moreover, Carney and Levin (2002) underline that pictures fulfill five traditional functions in text
processing: decorative, representational, organizational, interpretive, and transformational.
Decorative pictures simply decorate the page with little or no relation to the text content.
Representative pictures reflect some or all of the text content and are decisively the most widely used
type of illustration. Organizational pictures show qualitative relationships between different elements,
allowing a useful structural framework for text content. Interpretive pictures help clarify difficult texts
by providing the function of interpretation and reflection thanks to their explanatory aspect.
Transformative pictures contain systematic mnemonic (memory-enhancing) components designed to
enhance the reader’s recall by re-encoding text information to make it more tangible and then relating
it through a meaningful, interactive illustration. Similarly, Elia and Philippou (2004) propose four
functions (categories) of pictures in mathematical problem-solving: decorative, representational,
organizational, and informational. Decorative pictures do not provide any significant information
about the solution to the problem. Representative pictures illustrate all or part of the problem’s
content. Organizational pictures specify guidelines for drawing or written work that support the
solution procedure. Finally, informative pictures provide the information necessary to solve the
problem; in other words, the solution to the problem cannot be done without the picture.
It can be seen that there are both similarities and differences between the functions of non-textual
elements identified by different researchers (Carney & Levin, 2002; Duchastel & Waller, 1979; Elia &
Philippou, 2004; Kim, 2009, 2012). For example, the functions of pictorial illustrations by Duchastel
and Waller (1979) and Carney and Levin (2002) are useful in identifying the role of non-textual
elements in textbooks, especially in reading and science textbooks. Therefore, they may not be
sufficient to understand the functions of non-textual elements in mathematics textbooks. In contrast
to the typical non-textual elements in science, non-textual elements in mathematics are used not only
as informational tools, but also as tools for reasoning, argumentation, and reflection (Cuoco & Curcio,
2001). Based on this argument, Kim (2009, 2012) discussed the functions of non-textual elements in
mathematics textbooks under the headings of accuracy, connectivity, contextuality, conciseness, and
aesthetics. Although there are partial differences between all these functions of non-textual elements
mentioned by these researchers, it can be said that functions such as aesthetic and decorative,
descriptive, expressive, informative and interpretive, as well as logico-mathematical and connectivity
have many aspects that support and complement each other. Moreover, it is found that some
88 ● Genç, M., & Akıncı, M.
functions of non-textual elements such as decorative, representational, and organizational functions
are underlined by both Carney and Levin (2002) and Elia and Philippou (2004). Accordingly, the
conceptual framework of the study addresses all these highlighted similarities and differences in the
functions of non-textual elements together. Despite the significance of non-textual elements in
mathematics education, their function in the curriculum is still unclear to many teachers, especially
pre-service teachers. Thus, the purpose of this article is to examine how prospective teachers identify
the role of non-textual elements in a mathematics textbook. More clearly, it seeks to explore how pre-
service elementary mathematics teachers interpret the functions of non-textual elements from the
algebra learning domain in an eighth-grade mathematics textbook.
METHOD
Research Design
Qualitative studies are preferred in the research process, using unique methods to comprehensively
and in detail capture the phenomena under investigation (Creswell, 2017). An exploratory case study,
which is one of the qualitative research designs, deeply probes how individuals see themselves based
on their experiences, perceptions, and feelings depending on the context, and the reasons behind
them (Yin, 2014). Accordingly, this research study lends itself well to the use of a qualitative
exploratory case study that focuses on pre-service teachers’ interpretation of the non-textual elements
in mathematics textbooks for their intended use.
Participants
The study was conducted with a total of thirty-one pre-service elementary mathematics teachers
(twenty females and eleven males) enrolled in the third-year mathematics course “Analysis of
Mathematics Textbooks” at a public university in Turkey. This course is designed to provide students
with an overview of the pedagogical, structural, and organizational components of mathematics
textbooks including didactic and graphic visual design features, language standards, contribution to
meaningful learning, ease of use in the classroom, suitability for student-level, consistency with study
objectives, etc. The participants of the study also volunteered based on the convenience sampling
technique of the purposive (or purposeful) sampling method in qualitative research (Patton, 2014).
Data Collection
Data were collected through a task-based written questionnaire developed by the researchers and
administered to the pre-service teachers as an individual assignment to be completed outside of the
classroom. Evaluation done in this way not only gave the researchers flexibility in terms of time but
also provided more systematic and comparable data from the participants. This questionnaire includes
seven non-textual elements related to the learning outcomes in the algebra content area of the eighth-
grade mathematics textbook (Middle School and Imam hatip Middle School 8 Textbook by Böge and
Akıllı (2019)) published by the Ministry of National Education in Turkey. Pre-service teachers were
required to interpret the functions of twenty-three non-textual elements from the algebra content
area of this textbook. Among these elements, seven non-textual elements with various functions were
selected to be appropriate for the study. Another point considered in selecting these seven non-textual
elements is that they serve different outcomes of the algebra learning area in the mathematics
curriculum. Accordingly, one non-textual element for the learning outcome “Perform multiplication of
algebraic expressions” (Ministry of National Education [MoE], 2018), and two non-textual elements for
Volume: 13 Issue: 1 – Sakarya University Journal of Education ● 89
each of the learning outcomes “Explain identities with models” (MoE, 2018), “Solve first-degree
inequalities in one variable” (MoE, 2018), and “Explain the slope of a line using models, relate linear
equations and their graphs to slope” (MoE, 2018) were selected for investigation (see in Appendix 1).
Participants were asked to write down their thoughts about the usefulness of these non-textual
elements compared to symbols and textual information, what kind of roles or functions they have in
learning and teaching mathematics, and in which aspects they are more remarkable. Prior to data
collection, a task-based questionnaire was also piloted with four students not involved in the actual
study to assess the extent to which the questionnaire could elicit responses to address the research
question. Expert opinions were also taken into consideration. Based on the feedback received,
necessary amendments were made to arrive at the final version of the task-based written
questionnaire. In this respect, two questions were removed from the task-based questionnaire, which
initially consisted of five questions. One of them was excluded from the questionnaire because it was
a general statement covering other questions. The other one was excluded because it was thought
that pre-service teachers did not have sufficient expertise in textual and visual literacy about the lack
of features that non-textual elements should have. Therefore, it was not evaluated within the scope
of this study.
Data Analysis
Data analysis was performed using the content analysis method, which requires an in-depth analysis
of the collected data and allows to uncover of previously unspecified themes and dimensions (Corbin
& Strauss, 2015). All participants were coded from P1 to P31 to protect their confidentiality. Their
responses to the task-based questionnaire were received in writing. Qualitative data analysis software
NVivo (QSR International, 2012) was also used to assist in coding and categorizing data to identify
common themes and patterns. Using this software allowed us to eliminate potential conflicts in the
interpretation of data and ensure the accuracy of the research work by comparing it to the
participants’ responses. Taking into account both the relevant literature (Carney & Levin, 2002;
Duchastel and Waller, 1979; Elia & Philippou, 2004; Kim, 2009, 2012) and the data itself, an initial list
of codes was developed by assigning a code to each piece of information received from pre-service
teachers. For example, the preliminary code list for pre-service teachers’ interpretations of the image
of the fourth problem (see Figure 4 in Appendix 1), in which the algebraic identity
( )( )
22
a b a b a b− = − +
is shown by the area models, was created as attention, clarifying, embodying,
explanatory, facilitating, helping solution, informative, instructive, interpretative, logical, modeling,
rational, representative, simple, supportive, understandable, and useful. After the coding phase was
completed, similar codes were grouped into categories, which were further merged to form main
themes. For instance, codes such as embodying, modeling, and representative were included under
the theme of embodiment while codes such as facilitating, helping solutions, supportive,
understandable and useful were placed under the theme of organizing. Similarly, codes such as
clarifying, informative, and instructive were classified into the theme of informativeness while codes
such as interpretative, logical, and rational were grouped into the theme of reasoning. After the
creation of themes identifying the functions of the non-textual elements was completed, their
frequencies were also calculated. Participants’ responses were evaluated separately for each theme
created, and the ones that best reflected everyone’s views were quoted directly. Since credibility and
transferability were important to ensure the validity of the research, direct quotations with the views
of the participants were often included while presenting the findings, and the results were interpreted
based on those views. Through member checks, respondents were also asked to clarify or disagree
90 ● Genç, M., & Akıncı, M.
with something in the transcripts to identify and minimize potential bias. The aim was therefore to
disclose the research results accurately and impartially so that the emerging themes could form a
coherent and meaningful whole as much as possible (Merriam & Tisdell, 2016). Moreover, to facilitate
inter-coder reliability (Miles & Huberman, 1994), a mathematics education researcher was asked to
act as an external rater. Accordingly, in the last step of the analysis, in addition to the expert review,
this mathematics education researcher and the authors double-checked the codes in all transcripts
and reexamined the resulting categories. The comparison of the two codings resulted in an overall
agreement of 88 percent across all categories. The raters resolved any disagreements and revised the
codes until full agreement was reached for the categories to ensure the confirmability and
dependability of the data collected.
Ethical Principles
Ethics committee permission for this study was obtained from Zonguldak Bülent Ecevit University
Human Research Ethics Committee with the decision dated 05.10.2022 and numbered 221989/326.
FINDINGS
A cross-case analysis of the data from thirty-one pre-service teachers revealed ten themes that
captured their views on the functions of seven images selected from the eighth-grade mathematics
textbook in the algebra content area focusing on algebraic expressions and identities, linear equations,
and inequalities: (i) attractiveness, (ii) organizing, (iii) embodiment, (iv) informativeness, (v) reasoning,
(vi) conciseness, (vii) essentiality, (viii) decorativeness, (ix) contextuality, and (x) connectivity. Pre-
service teachers’ views on the functions of the non-textual elements were thematized and the
frequencies of each theme are given in Table 1. The following sections provide the findings concerning
these emergent themes.
Table 1
Frequencies and functions attributed by pre-service teachers to non-textual elements
Functions
Number of Participants
(Non-textual
Element-1)
(Non-textual
Element-2)
(Non-textual
Element-3)
(Non-textual
Element-4)
(Non-textual
Element-5)
(Non-textual
Element-6)
(Non-textual
Element-7)
Attractiveness
22 (71%)
20 (65%)
21 (68%)
20 (65%)
9 (29%)
18 (58%)
8 (26%)
Organizing
9 (29%)
21 (68%)
23 (74%)
22 (71%)
14 (45%)
27 (87%)
15 (48%)
Embodiment
17 (55%)
17 (55%)
20 (65%)
22 (71%)
8 (26%)
22 (71%)
12 (39%)
Informativeness
0 (0%)
8 (26%)
10 (32%)
8 (26%)
6 (19%)
19 (61%)
7 (23%)
Reasoning
4 (13%)
1 (3%)
3 (10%)
6 (19%)
8 (26%)
8 (26%)
1 (3%)
Conciseness
0 (0%)
0 (0%)
9 (29%)
5 (16%)
2 (6%)
0 (0%)
3 (10%)
Essentiality
0 (0%)
11 (35%)
1 (3%)
0 (0%)
9 (29%)
12 (39%)
8 (26%)
Decorativeness
21 (68%)
1 (3%)
0 (0%)
0 (0%)
1 (3%)
2 (6%)
0 (0%)
Contextuality
0 (0%)
1 (3%)
0 (0%)
0 (0%)
0 (0%)
0 (0%)
0 (0%)
Connectivity
0 (0%)
0 (0%)
0 (0%)
0 (0%)
1 (3%)
0 (0%)
0 (0%)
Volume: 13 Issue: 1 – Sakarya University Journal of Education ● 91
Attractiveness
Most of the pre-service teachers in this study mentioned the attractiveness of non-textual elements.
For example, twenty-two participants indicated that the image of the first problem about daily life (see
Figure 1 in Appendix 1), which requires writing a mathematical expression with first-degree
inequalities in one variable, can draw students’ attention to the problem. One of the pre-service
teachers put it as follows:
… with the picture given in a text, it was aimed to draw students’ attention to the problem by enabling
them to visualize the problem situation in their minds (P13).
Similarly, twenty-one participants stated that the image belonging to the third problem (see Figure 3
in Appendix 1), where the multiplication of algebraic expressions is illustrated using area models, has
the function of attractiveness and that it can arouse students’ curiosity and positively affect the
attempts to solve the problem. As one of the participants expressed:
... I think that it is a visual that will stimulate the curiosity and attention of the students and will capture
their interest in solving the problem (P27).
In addition, twenty participants pointed out that the attractive function of the image in the second
problem (see Figure 2 in Appendix 1), dealing with transforming the real-life problem into a
mathematical statement, can increase students’ motivation to solve the problem. As stated by one of
the participants:
…This picture has a role in raising the motivation of students...The fact that the visual is colorful and
the tools used in real life may attract students’ attention (P9).
Organizing
Organizing is a function emphasized by most of the pre-service teachers in all non-textual elements.
For instance, twenty-seven participants referred to the organizing function of the image in the sixth
problem (see Figure 6 in Appendix 1), where the slope is explained with models from everyday life.
They mentioned that this image supports textual information and makes it useful for solving the
problem. One of the participants expressed this as follows:
…I think textual information about the problem is supported by this picture and it is quite useful for
solving the problem. … It is a visual that helps students understand the problem and therefore
reinforces the topic being covered (P28).
Twenty-two participants also considered the purpose of using the image belonging to the fourth
problem (see Figure 4 in Appendix 1), in which identities are explained with area models, as organizing
because the image facilitates the solution of the given problem by organizing given information
logically and coherently to produce mathematical ideas. As stated by one pre-service teacher:
…this image organizes information given in the problem and facilitates its solution in a way that
encourages students to form their own ideas (P10).
Similarly, twenty-one participants mentioned the organizing function of the image used for the second
problem (see Figure 2 in Appendix 1), which is about expressing the real-life problem of inequalities
mathematically, as it provides students with information about the problem and helps them plan its
solution. As one of the participants remarked:
92 ● Genç, M., & Akıncı, M.
… equal arm weighing scales in the figure are used to inform students, support their understanding,
and facilitate the planning of the solution for the problem (P15).
Embodiment
Embodiment is also a function revealed by most of the pre-service teachers in all non-textual elements.
For example, twenty-two participants expressed their thoughts about the role of the image of the
fourth problem (see Figure 4 in Appendix 1), in which identities are illustrated with area models. Pre-
service teachers mentioned that the given image allows students to discover the algebraic identity
( )( )
22
a b a b a b− = − +
by embodying the learning process. They stated that mathematical principles
that are difficult to visualize in a way that makes sense in the mind are embodied using visual
representations or images. That is:
…thanks to this visual, the identity
( )( )
22
a b a b a b− = − +
, which is an abstract fact for students, is
modeled using areas, and thus it is proved by embodied (P24).
Likewise, twenty participants argued that the image belonging to the third problem (see Figure 3 in
Appendix 1), which is about performing multiplication with algebraic expressions with area models,
embodies the given problem and makes its solution understandable. As stated by one of the
participants:
…the geometric shapes in the picture are visual elements used to make algebraic expressions
meaningful. Explaining this subject only with algebraic expressions remains abstract for students, but
the shapes used to represent the problem embody its solution for students (P13).
Also, twelve participants explained that the image of the seventh problem (see Figure 7 in Appendix
1), discovering the slopes of parallel and perpendicular lines, embodies abstract concepts in the
problem that are difficult for students to understand. One of the participants put it simply:
…If only textual information is included, the problem may be difficult to understand and the problem
may remain abstract. The inclusion of the image embodies the solution in the mind (P23).
Informativeness
Informativeness is another function that is noted by the pre-service teachers in all images except the
image of the first problem. For example, nineteen participants mentioned the informative function of
the image on the sixth problem (see Figure 6 in Appendix 1), illustrating that the slope is the ratio of
vertical length to horizontal length in a model related to daily life. As pointed out by one participant:
…The three shapes with different slopes given in the image make it easier for students to understand
that as the height increases, the slope will increase and as the length increases, it will decrease. For
this reason, the images provided are informative to understand the subject and contribute to learning
(P17).
Eight participants also supported that the image (see Figure 4 in Appendix 1), in which the algebraic
identity
( )( )
22
a b a b a b− = − +
is proved by the area models, has an informative function as it makes
the learning more effective and permanent by explaining the relationships between concepts. One of
the pre-service teachers expressed that:
Volume: 13 Issue: 1 – Sakarya University Journal of Education ● 93
…demonstrating the identity with an area model instead of just showing it as a long mathematical
expression makes learning more permanent. Even if students forget the algebraic identity, they can
remember this model and prove it themselves (P5).
Similarly, seven participants emphasized the informative function of the image belonging to the
seventh problem (see Figure 7 in Appendix 1), examining the relationship between parallel and
perpendicular lines and the concept of slope. As one participant stressed:
…this visual is used to explain the relationship between the slopes of two lines that are parallel or
perpendicular… In place of writing a long mathematical text, it is used to summarize the text and
highlight the important information students need to know (P10).
Reasoning
The other function of the non-textual elements stated by the pre-service teachers is reasoning. For
example, eight participants expressed that allowing the comparison of the slopes given in the figure
on the sixth problem (see Figure 6 in Appendix 1) contributes to the interpretation of factors affecting
the magnitude of the slope, thus revealing the reasoning function. As put by one of the participants:
…thanks to the shapes in the given figure, it can be easily interpreted in which cases the slope increases
and in which cases it decreases (P12).
Similarly, six participants mentioned that the image of the fourth problem (see Figure 4 in Appendix
1), in which the algebraic identity
( )( )
22
a b a b a b− = − +
is shown by the area models, allows the
interpretation of the problem situation and reasoning by thinking deeply about the problem situation.
In other words, they stated that the visual can encourage students to think mathematically, as it helps
to see how the given problem can be solved. In the words of a pre-service teacher:
…instead of memorizing the given identity, the students try to think and interpret the reason for it with
the help of the given visual...if only textual information is included, it will be more abstract and difficult
to explain and interpret the identity (P28).
Conciseness
Conciseness is another function underlined by the pre-service teachers in all non-textual elements
except the first, second, and sixth non-textual elements. For instance, nine participants stated that the
image presented in the third problem (see Figure 3 in Appendix 1) about performing the multiplication
of algebraic expressions can simply and concisely convey the meaning of the multiplication operation
on algebraic expressions by modeling the equation
( )
2
2 1 2x x x x+ = +
using areas. As one of the
participants noted:
…I think it is very helpful compared to textual information because what is meant to be explained in
the text is modeled with a given visual more simply and understandably (P26).
In a like manner, three participants stated that the image of the seventh problem (see Figure 7 in
Appendix 1), aiming to relate the concept of slope to parallel and perpendicular lines, shows the
relationship between these concepts clearly. As posited by one of the participants:
…It is a useful visual as it clearly shows what is meant to be given in the problem (P11).
94 ● Genç, M., & Akıncı, M.
Essentiality
Essentiality was indicated as another function of non-textual elements in all visuals except the first and
fourth ones. For example, eleven participants pointed to the essentiality function by stating that the
mathematical expression required in the problem could not be written without the image given in the
second problem (see Figure 2 in Appendix 1), focusing on expressing mathematical statements of real-
life problems related to inequalities. As remarked by one of the participants:
…the visual is necessary for solving the problem. Otherwise, the text describing the question is useless
without the image provided (P4).
Similarly, nine participants stated that the image of the fifth problem (see Figure 5 in Appendix 1), in
which the algebraic identity
( )
222
2a b a ab b+ = + +
is shown, is essential for understanding the related
problem. In other words,
…the given visual is part of the problem and is necessary for the solution of the problem (P1).
Decorativeness
Most of the pre-service teachers emphasized the decorative function of the given non-textual
elements. For instance, twenty-one participants pointed to the decorativeness of the visual given with
the first problem (see Figure 1 in Appendix 1), which is about writing a mathematical expression
suitable for daily life situations involving first-order inequalities with one unknown. They also
commented on whether the given problem can be solved without its visuals. One of the participants
put it as follows:
…this visual does not contain any information about the solution to the problem, it is only used for
decorative purposes. Even if the visual is not used here, this problem can be solved (P10).
Contextuality
Contextuality was highlighted by one pre-service teacher in the image of the second problem (see
Figure 2 in Appendix 1), which requires transforming a real-life problem based on inequalities into
mathematical expressions. It was assumed that with this image, students can establish a relationship
between everyday life and mathematics. As stated by one of the pre-service teachers:
…given a weighing instrument used in everyday life, this visual helps students to relate the subject of
inequality to daily life (P3).
Connectivity
Connectivity emerged as another function of non-textual elements, which was pointed out by only one
participant in the image of the fifth problem (see Figure 5 in Appendix 1), where the algebraic identity
( )
222
2a b a ab b+ = + +
is proved. A pre-service teacher asserted that since the given image is related
to the content of the problem there is a connectivity function.
…I think that the visual chosen in connection with the textual information of the problem helps
students embody what is described in the text and visualize the problem in their minds (P1).
Volume: 13 Issue: 1 – Sakarya University Journal of Education ● 95
DISCUSSIONS AND CONCLUSIONS
In this study, pre-service elementary mathematics teachers emphasized various functions of seven
non-textual elements that are part of algebra learning outcomes in an eighth-grade mathematics
textbook. In this regard, ten different functions were identified in total: (i) attractiveness, (ii)
organizing, (iii) embodiment, (iv) informativeness, (v) reasoning, (vi) conciseness, (vii) essentiality, (viii)
decorativeness, (ix) contextuality, and (x) connectivity. It was seen that pre-service teachers attributed
at least five different functions to each of the non-textual elements (see Table 1). Therefore, as stated
in the literature, it has been shown that non-textual elements can have more than one function, or
more than one meaning can be attributed to a non-textual element (Carney & Levin, 2002; Elia &
Philippou, 2004; Kim, 2009, 2012; Lee & Ligocki, 2020).
It has often been emphasized that each non-textual element used in this study has the functions of
attractiveness, organizing, and embodiment. Similarly, some studies asserted that most of the pictures
used in mathematics textbooks were attractive because pictures did contribute to the attractiveness
of the learning material and the enjoyment of reading (Biron, 2006; Peeck, 1993). Besides, in the
studies analyzing the functions of the pictures in the problem-solving process (Elia & Philippou, 2004),
the roles of pictorial illustrations in mathematics textbooks (Carney & Levin, 2002), and the way
prospective teachers interpret and use non-textual elements in mathematics curriculum materials (Lee
& Ligocki, 2020), visuals were found to organize the problem-solving process. In addition, visualization
is essential for mathematical generalization and abstraction because visual representations can
embody a concept in various ways and make it comprehensible (Demircioğlu & Polat, 2015; Dufour-
Janvier, Bednarz & Belanger, 1987; Yilmaz & Argun, 2018). Therefore, it can be argued that these three
functions of visuals (attractiveness, organizing, and embodiment) will have a critical role in the
effective internalization of the abstract content of algebra in mathematics textbooks.
Another important finding is that reasoning, which pre-service teachers believe plays a crucial role in
the implementation of all mathematical skills, is a function expressed in all non-textual elements used
in the study. It has long been known that using certain types of representations (visual, concrete, etc.)
allows students to develop mathematical skills such as reasoning and helps them acquire advanced
problem-solving skills (Presmeg, 2020). Therefore, considering the fact that non-textual elements used
in teaching mathematics facilitate students’ reasoning processes (Alsina & Nelseni, 2006), it is
important to include such non-textual elements in the algebra content area of mathematics textbooks.
It was also found that pre-service teachers attributed an informative function to six non-textual
elements and an essentiality function to five non-textual elements in the study. Seffah (2017) indicated
that in most textbooks, visual representations, which are an important tool that can contribute to a
comprehensive understanding of the concept of the series, are rarely used, and they have the function
of essentiality as in this study. Karakaya (2011) noted that most of the visual representations of
functions used in mathematics textbooks have an informative function. Therefore, it is important for
the teaching of algebraic concepts that the non-textual elements in the textbooks have informative
and essentiality functions, as they initiate mathematical thinking by explaining the text and providing
the necessary information for solving the problem.
Moreover, conciseness, interpreted as mathematical simplicity by pre-service teachers, was a function
that emerged in the four non-textual elements used in the study. Pettersson (2001) contended that
too much detail or complexity reduces interest in visual content, while too little detail or complexity
makes it impossible to understand the picture. In other words, since mathematics is such a precise
96 ● Genç, M., & Akıncı, M.
subject, the ambiguity of complex visual representations can prevent students’ understanding of the
concept (Goldin & Shteingold, 2001). Simple or concise non-textual elements facilitate understanding
of the mathematical concept and making connections between the concept and other related
mathematical concepts (Kim, 2012). Because concise non-textual elements can clearly and effectively
convey the meaning and idea when teaching a new concept, such elements can also help students
better understand algebraic concepts.
Another function expressed in four non-textual elements is decorativeness, referred to as the
aesthetics or visual appeal of the images. Sinclair (2006) argued that aesthetics should be considered
an important step for success in mathematics because it is closely related to students’ understanding
and learning styles of mathematics. Goldin (2000) emphasized that aesthetics is important not only to
help students discover beauty in mathematics, but also because it affects students’ affective
characteristics such as emotions, beliefs, and attitudes toward mathematics. In this respect, one can
better understand that the decorative function of the visuals in the textbooks is more valuable for
teaching algebra.
On the other hand, contextuality, mentioned as the representation of mathematical expressions in a
realistic context, was a function emphasized by the pre-service teacher only in one non-textual
element. Wiggins (1993) asserted that it is inappropriate to evaluate information out of its context.
Ferratti and Okolo (1996) showed that students’ thinking skills and attitudes improve as they solve
problems in realistic contexts. Non-textual elements with a contextual function can facilitate students’
understanding and learning through connections between mathematics and real life (Kim, 2012).
Therefore, it should be kept in mind that such non-textual elements in textbooks can enable students
to understand and deepen algebraic concepts by encouraging them to think in their own context. In
addition, connectivity, interpreted by the pre-service teacher as the non-textual elements being closely
related to the mathematical content in the texts, was a function specified only in one non-textual
element, as in the contextuality function. Since visual representations can serve as models or problem-
solving tools to show students what they cannot see in texts and symbols (Arcavi, 2003), it is important
to establish close connections between mathematical texts and visual representations in teaching
algebra. Because textual literacy may not be at the same level as visual literacy for every student,
students who have difficulty reading and comprehending mathematical texts can learn more and
understand better through visuals (Kim, 2012). Therefore, visual representations can provide some
students with opportunities to learn mathematics that they cannot understand only from algebraic
textual expressions.
Overall, this study provides an overview of pre-service elementary mathematics teachers’
interpretations of the functions of non-textual elements from the content area of algebra in a
mathematics textbook. Pre-service teachers were found to have diverse but sometimes overlapping
interpretations of the functions of each non-textual element. However, the functional diversity of non-
textual elements may have challenged their interpretations, as visual processing skills and strategies
are required to interpret the intended use of non-textual elements. Some pre-service teachers had
difficulty seeing the connections between the information in the pictures and the mathematical texts
containing algebraic expressions. Understanding the intended message of the non-textual elements in
mathematics textbooks seems to be something that some prospective teachers are unlikely to do
unless they have carefully planned, intentional training on the matter. Otherwise, despite increasing
informational texts and visual content in textbooks, these prospective teachers will hardly be able to
teach their students to read visual elements as well (Metros, 2008). Therefore, in order to enhance
Volume: 13 Issue: 1 – Sakarya University Journal of Education ● 97
pre-service teachers’ understanding of the functions of non-textual elements, not only should
researchers develop various instructional approaches that promote pre-service teachers’ visual
literacy, but these instructional approaches to visual literacy should also be incorporated into teacher
education and professional development. Moreover, while our study will guide future research by
documenting the functions of non-textual elements encountered in an algebra content area of a
mathematics textbook, it would be valuable to extend this work to other possible functions of non-
textual elements by considering different learning domains and various mathematics textbooks.
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Author Contributions
The authors contributed equally for writing the article, conceptualization of the article, data collection,
analysis and discussion.
Conflict of Interest
No potential conflict of interest was declared by the author.
Supporting Individuals or Organizations
No grants were received from any public, private or non-profit organizations for this research.
Ethical Approval and Participant Consent
Ethics committee permission for this study was obtained from Zonguldak Bülent Ecevit University Human
Research Ethics Committee with the decision dated 05.10.2022 and numbered 221989/326.
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