Some Conceptual Metaphors for Rational Numbers as Fractions in Swedish Mathematics Textbooks for Elementary Education

ABSTRACT

This study examines the presence of three conceptual metaphors for fractions – The measuring stick metaphor, Arithmetic is motion along a path, and Arithmetic is object construction – in four common and popular Swedish mathematics textbook series for years 1–3. I analyse the introduction of fractions and the kinds of tasks students are given in these books. The results show an abundance of labelling exercises related to Arithmetic is object construction, with representations of geometric shapes, in three of the book series. One book series reversely introduces fractions with The measuring stick metaphor and Arithmetic is motion along a path, with a focus on number line representations. The consequences of these variations in fraction introduction and treatment are discussed in relation to previous research and to process-object theories.

KEYWORDS:

• Textbook analysis
• fractions
• metaphor
• conceptual metaphor theory
• process-object

Introduction

In recent decades, mathematics education has presented metaphor as an important concept, describing the grounded processes by which mathematical concepts are constituted and understood (Lakoff & Núñez, 2000; Soto-Andrade, 2014). From the viewpoint of mathematics as subject matter or a discipline (Schiralli & Sinclair, 2003), conceptual metaphors (CM) are seen as the genesis of mathematical ideas (Núñez, 2008). Conceptual metaphor theory (CMT) proposes that basic mathematical concepts are constituted through embodied experiences in a source domain, which are metaphorically mapped onto numbers and simple arithmetic in a target domain (Lakoff & Núñez, 2000). From this perspective, fractions (in this paper, the term “fraction” denotes both the rational number concept, and the notation, ${\displaystyle \frac{a}{b}}$, of a rational number in fractional form) are constituted through three CMs: The measuring stick metaphor (TMSM), Arithmetic is motion along a path (AMAP), and Arithmetic is object construction (AOC) (Lakoff & Núñez, 2000). This perspective on metaphors will be further elucidated in the theoretical background.

This metaphorical move is also seen in the inherent duality of the fraction concept, as both process and object (Sfard, 1991). In different degrees of abstraction, fractions can be seen both as a process on concrete objects, for instance, the partitioning of wholes into thirds, and as the higher level object of ${\displaystyle \frac{1}{3}}$, with which certain operations can be executed. Solving fraction tasks requiring operations with the symbolic fraction notation as a reified object (Sfard, 1991) is more difficult for students than tasks based on the process dimension of fractions, for example, tasks with visual representations such as partitioned geometric shapes (Pantziara & Philippou, 2012).

Rational number concepts are among the most complex and important mathematical ideas students encounter (Behr et al., 1983). Knowledge of fractions is important for mathematical ability and as a key component for students’ number sense (Bailey et al., 2012). In Sweden, fractions are introduced in the syllabus for years 1–3, with an emphasis on fractions as part-whole, naming simple fractions and understanding their relationship to the integers (Swedish National Agency for Education, 2018).

The 2014 and 2015 results from the Swedish national tests in mathematics for year 6 show that only 34% of the students managed to compare the sizes of ${\displaystyle \frac{3}{7}}$ and ${\displaystyle \frac{5}{9}}$ correctly, and 38% of the students calculated 1 – ${\displaystyle \frac{1}{5}}$ correctly (Pettersson et al., 2014; Pettersson & Thisted, 2015). Similar difficulties have been known for a long time in research (see Kerslake, 1986) and there are several misconceptions concerning the magnitude of fractions (Stafylidou & Vosniadou, 2004).

In Sweden, the teaching of mathematics is strongly governed by textbooks (Swedish Schools Inspectorate, 2009) and 90% of Swedish mathematics teachers report that textbooks serve as a basis for their teaching (Mullis et al., 2012). No substantial research exists that shows when and how fractions generally are introduced in Swedish schools, but it can be assumed that early fraction instruction largely focuses on fractions as part-whole since this sub-construct is clearly connected to children’s everyday experiences of partitioning (Ahl & Helenius, 2018; Empson, 1999). Although part-whole is one aspect of an understanding of fractions and a common introduction in textbooks internationally (Lee et al., 2021), research points to the limitation in introducing fractions as part-whole (Simon et al., 2018), and to the importance of students encountering different interpretations of fractions (Lamon, 2012). From a conceptual change perspective, learning new information can cause conflicts with prior knowledge based on everyday experiences (Vamvakoussi & Vosniadou, 2004). Further, textbooks are often structured around an idea of progression in topic complexity and developmental complexity (Schmidt et al., 1996). This progression, order, and manner of presentation of different fraction concepts in textbooks for young students may be important for their later fraction learning.

This paper examines when and how the three CMs concerning fractions, AOC, TMSM and AMAP, as manifested through three types of visual representation – partitioned geometric shapes, everyday objects such as pizzas and cakes, and the number line – are presented in Swedish mathematics textbooks for students in years 1–3. The analysis examines the textbook tasks connected to the three CMs and the type of responses required by the students. The analysis is guided by the following research questions:

1. When are the three fractions-related CMs, with their visual representations, introduced in Swedish mathematics textbooks for years 1–3?

2. What is the prevalence of these three CMs across the analysed textbooks?

3. What kind of responses on fraction tasks connected to these CMs are required from students in these textbooks?

Previous Research

Literature relevant for this study includes research regarding the treatment of fractions in textbooks; two well-established interpretations of rational numbers as fractions – part-whole and measure (Kieren, 1980); and some important aspects of knowledge of fractions for students in elementary education, namely partitioning, labelling, comparison and equivalence, unitising and arithmetical operations.

Fractions as Part-Whole

The part-whole sub-construct is probably the most common interpretation of fractions in school and often the first that students encounter (Lee et al., 2021). Exercises with equal sharing, partitioning, labelling parts as fractions, and unitising – the process of operating on the unit, for instance by re-partitioning and putting together smaller parts into “chunks” – are principal factors for students’ development of basic fraction conceptions (Empson, 1999; Lamon, 1996; Pantziara & Philippou, 2012; Westenskow & Moyer-Packenham, 2015). Representations of geometric shapes are powerful tools for comparing relative sizes of fractions, and for the addition and subtraction of fractions (Cramer et al., 2008).

It is also well established that the part-whole interpretation with area representations of fractions is insufficient (Lamon, 2012), and an overly rigid connection to this can hinder the development of other interpretations of fractions (Pitkethly & Hunting, 1996). To fully understand the sub-construct, knowledge of several aspects are required – that the whole must be divided into parts of equal size (Lamon, 2012), that the parts taken together must exhaust the whole (Charalambous et al., 2010) and “that the relationship between the parts and the whole is conserved, regardless of the size, shape, arrangement, or orientation of the equivalent parts” (p. 296), etc.

Fractions as Measure

A common representation of the measure sub-construct is the number line. The ability to compare the magnitude of fractions using a number line in early education is positively related to later success in mathematics (Resnick et al., 2016) and the number line supports students’ understanding of relations between fractions (Sidney et al., 2019). Researchers emphasise the importance of early exercises with number lines that go beyond 1, as well as the construction of number lines with successive partitioning and repartitioning, finding equivalent fractions, comparing fractions and finding fractions between two given fractions (Lamon, 2012; Novillis Larsson, 1980; Resnick et al., 2016).

However, students can find the number line difficult and may find it easier to express fractions using area representations (Novillis Larsson, 1980). Several researchers have reported how students find it difficult to correctly place fractions on a number line (Witherspoon, 2019; Zhang et al., 2015). A common misconception among some students is that fractions are numbers that only exist between 0 and 1 (Resnick et al., 2016). Some researchers believe that these difficulties can be traced to, among other things, the privileged place of the part-whole sub-construct in instruction (Ahl & Helenius, 2018), but also that the number line requires a complement of symbols to be fully usable (Bright et al., 1988).

Textbooks

International and Swedish research show that textbooks play a significant role in teachers’ lesson planning (Huntley, 2008; Jablonka & Johansson, 2010; Johansson, 2017; Usiskin, 2013). Textbooks control material selection and teachers often follow the content and order as presented in the textbook (Brehmer et al., 2016; Johansson, 2006; Nicol & Crespo, 2006; Pepin & Haggarty, 2001). Topics from the textbook are likely to be introduced in the classroom, whilst topics not included are most likely not presented by the teacher (Johansson, 2006).

In a Nordic context, several textbook studies concern materials for older students – Icelandic textbooks aimed at year 8 (Halldórsdóttir, 2017), textbooks for Swedish university students (Lithner, 2004) or Finnish textbooks for upper secondary school (Viholainen et al., 2017) – whilst little is known about how mathematical ideas are presented in books aimed at young children (Sayers et al., 2021).

Few studies specifically examine how fractions are treated in textbooks. Some studies focus on the addition and subtraction of fractions (Hwang et al., 2021), the multiplication of fractions (Kar et al., 2018) and the division of fractions (Bütüner, 2020; Li et al., 2009). Other studies focus on how real-life contexts are used in fraction problems, and the degree to which fraction tasks demand conceptual or procedural knowledge (e.g., Yang et al., 2010).

In international research, the part-whole sub-construct comes across as the most common introduction of fractions in textbooks, spread out over the first three years of instruction, as seen in textbooks from the USA, Kuwait and Japan (Alajmi, 2012), Japan, Korea and Taiwan (Watanabe et al., 2017), Singapore and South Korea (Lee et al., 2021) and Indonesia (Rahmawati et al., 2020). In a majority of these textbooks, fractions as measure are introduced in years 3 and 4. Common fraction tasks in elementary school textbooks involve labelling (e.g., writing “${\displaystyle \frac{1}{3}}$” next to a partitioned circle), pointing out (e.g., encircling a certain fraction), comparing (e.g., by drawing lines connecting equivalent fractions), colouring different geometric shapes (e.g., colouring one part of a partitioned square) as well as word problems (Alajmi, 2012; Daud, 2020; Lee et al., 2021).

Theoretical Background

A concept originating from the domain of cognitive linguistics, a CM is understood as a “grounded, inference-preserving cross-domain mapping” (Lakoff & Núñez, 2000, p. 6), a cognitive structure that allows us to use a source domain, often concrete and physical, to reason about another, often abstract, target domain. Fractions within CMT are constituted through three CMs – AOC, TMSM and AMAP (Lakoff & Núñez, 2000). These are embodied ideas constitutive of the mathematical phenomenon and the conceptual basis for fractions. Embodiment is the understanding that our bodies, brains and “everyday functioning in the world structures human concepts and human reason” (p. 5). For instance, AOC is a conceptual structure based on the everyday experience of manipulating groups of objects as the basis for simple arithmetic. In this study, the concept of CM is understood as one important part for the constitution of the theoretical idea of fractions, with educational implications.

Further, these CMs are visible in two of Kieren’s (1980) sub-constructs for fractions focused on in this study. In the same manner as the sub-construct of part-whole, fractions are metaphorically understood within AOC as the splitting and fitting together of parts of unit objects (Table 1). Through TMSM on the other hand, fractions are metaphorically understood as physical segments (numbers) that are split into shorter segments of equal size to form simple fractions and fitted together to form complex fractions (Table 2). AMAP metaphorically constructs numbers as points on a path, and moving along this path in different directions as arithmetical operations (Table 3). In this context, these two CMs are regarded as equivalent to the sub-construct of fractions as measure.

Table 1. Fractions based on Arithmetic is object construction (Lakoff & Núñez, 2000).

A part of a unit object (made by splitting a unit object into n parts)	→	A simple fraction (1/n)
An object made by fitting together m parts of size 1/n	→	A complex fraction (m/n)

Table 2. Fractions based on the measuring stick metaphor (Lakoff & Núñez, 2000).

A part of a physical segment (made by splitting a single physical segment into n equal parts)	→	A simple fraction (1/n)
A physical segment made by fitting together (end-to-end) m parts of size 1/n	→	A complex fraction (m/n)

Table 3. Fractions based on Arithmetic is motion along a path (Lakoff & Núñez, 2000).

Starting at 1, find a distance d such that by moving distance d toward the origin repeatedly n times, you will reach the origin. 1/n is the point-location at distance d from the origin,	→	A simple fraction (1/n)
Point-location reached moving from the origin a distance 1/n repeatedly m times.	→	A complex fraction (m/n)

One term often used in mathematics education research to describe how fractions can be concretised, is model. Area models and linear models (Sidney et al., 2019) are common terms. A basic assumption in this text is that behind each model there is a metaphor, and in order to create a mathematical model we need an interconnection between two conceptual domains – an interactive linking only possible in the presence of a metaphor (Carreira, 2001). In such an interconnection of the form x is y, the model consists of the properties of x which metaphorically mediate the phenomenon y (Carreira & Lerman, 2006). A model clarifies and concretises a metaphorical connection (Bonderup Dohn, 2018), and is visually represented by, for instance, pictures, written text, or written symbols, often referring to real-world situations.

Fractions as Process and Object

A key premise of this study is that abstract notions such as fractions present a duality. Fractions can be understood both structurally – part object – and operationally – part process (Sfard, 1991). This duality of fractions and the notation ${\displaystyle \frac{a}{b}}$, indicates both an embodied process, for instance, the partitioning of a whole, but also a disembodied abstract object, a static structure with which calculations can be performed. These two understandings can be understood as complementary and essential to concept formation. Sfard (1991) emphasises that a profound insight into the processes underlying mathematical concepts should be viewed as a basis for understanding such concepts. According to Pantziara and Philippou (2012), the process of abstraction – in the context of fraction instruction – starts when the students’ attention shifts from different sharing situations to the outcome as an object signified by a fraction notation (${\displaystyle \frac{a}{b}}$).

The process of concept formation moves through three stages: interiorisation, where a learner performs operations on lower-level mathematical objects; condensation, where lengthy sequences of operations are “squeezed” into more manageable units, and reification, where the full process is solidified into an object, a static structure (Sfard, 1991). In this process, names, symbols, graphs, and other visual representations play a vital role, since such a static object-like representation, “squeezes the operational information into a compact whole” (p. 26).

Pantziara and Philippou (2012) developed several fraction tasks (aimed at year 5 students) as operationalisations of these three stages of concept formation. At an interiorisation stage, common tasks were the counting of shape-denominators and the shaded parts-numerator and finding the fraction or placing a fraction on a number line with markings equal to the denominator. At a condensation stage, tasks included the partitioning of already-partitioned wholes and finding the fraction, and the placing of a fraction represented by a partitioned and shaded rectangle on a number line. Finally, the reification stage saw tasks such as partitioning already-partitioned wholes into parts of unequal size and shape, tasks with unitising actions, as well as tasks naming the fraction ${\displaystyle \frac{1}{6}}$ between 0 and ${\displaystyle \frac{1}{2}}$ on a number line partitioned into three segments of equal size.

These steps in the concept formation process, with operationalised tasks, will be highlighted in the discussion in this study, even though no analysis is made based on these concepts.

Research Design

In Sweden, textbooks are published by private organisations in the form of companies or foundations. No central regulations exist for the control or publication of textbooks, and most textbooks are created by teachers and researchers, through the publishers (SOU 2021:70). The sample of textbooks was based both on convenience and the fact that these are used widely in Swedish schools. The books are published by four of Sweden’s leading publishers: Sanoma Utbildning AB, Majema, Gleerups and Studentlitteratur AB. These publishers each have several different mathematics textbook series. To limit the scope of this study, a selection has been made of books aimed at years 1–3. The textbook series are Matte Direkt Triumf (Bergwik & Falck, 2020, 2021) (MDT), Mondo Matematik (Brorsson, 2017) (MM), Mitt i prick (Mårtensson & Öhman, 2016, 2018) (MiP), and Favoritmatematik (Karppinen et al., 2018; Ristola et al., 2012) (FM). All series consist of six books per series, where book 1A is intended to be used in the first semester of the first year, book 1B in the second semester of the first year, book 2A in the first semester of the second year, and so on. This generated 24 books eligible for the study.

The books in the series that, based on the research questions, did not contain any data relevant to the study, were excluded. This meant excluding 13 books that did not contain any one of the three CMs with their visual representations of geometric shapes, everyday objects and number lines, leaving 11 books to be part of the study. No teacher guides or adjacent teaching materials were included. This was done to limit the scope of the study.

Coding and Unit of Analysis

This study uses deductive content analysis as a research method, structured around previous research and theories with the aim of systematically describing and quantifying phenomena (Elo & Kyngäs, 2007; Krippendorff, 2004; Mayring, 2000); in this case textbook tasks connected to two fraction sub-constructs and three CMs, and the student responses required on these tasks. Answering the research questions required both a quantitative and a qualitative content analysis.

To conduct the analysis, a framework was developed around the three CMs for fractions. These CMs are in turn the basis for area and linear models, and manifested through three types of visual representations: geometric shapes, everyday objects, and the number line. The AOC is manifested as images of geometric shapes, and as images of pizzas, cakes, and so on. The CMs, TMSM and AMAP are manifested through the number line, often starting with 0 and ending with 1, or going past 1.

Although the idea of fractions as discrete objects is part of the part-whole sub-construct and common in textbooks, this interpretation was excluded from the framework, to limit the scope of the study. Also excluded were one common type of task, namely, the counting of parts of a partitioned unit object not requiring a fraction answer (Lamon, 2012).

With respect to the CMs, models and visual representations, a categorisation was made based on the previously mentioned aspects of fraction tasks and knowledge common in mathematics textbooks, and in relevant literature concerning fractions (e.g., Behr et al., 1983; Bezuk & Cramer, 1989; Lamon, 2012; Steffe & Olive, 2010; Tunç-Pekkan, 2015). These aspects constituted five categories: partitioning, labelling, arithmetical operations, comparing and equivalence, and unitising. Using a deductive category application (Mayring, 2000), Table 4 describes the analysis framework with five categories of fraction aspects (A to E) and six subcategories of type of response connected to TMSM/AMAP and five categories (F to J) with six subcategories connected to AOC. Hence, the unit of analysis is the distinct types of student response required in a task.

Table 4. Analysis framework.

Conceptual metaphor	Model	Representation	Category: Fraction aspect	Subcategory: Type of response
TMSM/ AMAP	Linear	Number line	A. Partitioning	Partitioning/creating number lines
			B. Labelling	B1. Labelling/placing simple/complex fractions on number lines between 0 and 1
				B2. Labelling/placing mixed numbers
			C. Operations	Arithmetical operations
			D. Comparing	Comparing/finding equivalent fractions
			E. Unitising	Partitioning/creating new subdivisions and units
AOC	Area	Geometric shapes and everyday objects	F. Partitioning	Partitioning/creating unit objects
			G. Labelling	G1. Labelling/selecting/colouring one/several parts as a simple/complex fraction
				G2. Labelling/selecting/colouring parts as mixed numbers
			H. Operations	Arithmetical operations
			I. Comparing	Comparing/finding equivalent fractions
			J. Unitising	Partitioning/creating new “chunks”/units

In the first step of the analysis, a page count was conducted, and the total number of pages was recorded. Then the first and subsequent fraction chapters were located, and the first instance and introduction of any of the three CMs within these chapters and their representations noted. This generated Table 5, which shows the number of pages in the respective book series as a whole and in relation to this, on which page the respective CMs were introduced. The relevance of these data is based on two assumptions: (1) that teachers follow the sequence of the textbook content (Brehmer et al., 2016), and (2) that there exists a progression in topic complexity and developmental complexity in textbooks (Schmidt et al., 1996) – that is, the general pattern of basic topics is treated early in textbooks, and more complex topics presented later. Table 5 shows the sequencing and dispersion of the CMs, thus giving an indication of different fraction concepts’ perceived complexity.

Table 5. Book and pages with first instance of AOC and TMSM/AMAP.

		AOC		TMSM/AMAP
Textbook series	No. of pages	Book	Page	Book	Page
MDT	960	2A	444	3A	724
MM	960	2B	546	2B	542
MiP	924	2A	418	–	–
FM	1264	2A	586	3B	1059

In a second step, all responses within a task concerning the CMs were counted. A task in a book could require several responses from a student. For instance, a task with a number line, and adjacent three empty boxes required three responses with respect to placing fractions, producing three units of data related to TMSM/AMAP and subcategory B1. Conversely, the colouring of, for instance, 9 parts of 10 in a partitioned square counted as one response, producing one unit of data related to AOC and subcategory G. This analysis generated Table 6.

Table 6. Responses connected to subcategories within TMSM/AMAP and AOC.

Book	TMSM/AMAP							AOC							Both
	A	B1	B2	C	D	E^f	Tot.	F	G1	G2	H	I	J	Tot.		Tot.
MDT 2A^a								21	59	0	0	0	0		0
MDT 2B^b								1	55	0	0	0	0		0
MDT 3A	0	7	0	0	0			0	98	0	22	0	0		8
MDT 3B	0	8	0	0	0			4	30	0	6	0	0		0
Tot.	0	15	0	0	0	0	15	26	242	0	28	0	0	296	8	319
%	0	4.7	0	0	0	0	4.7	8.2	75.9	0	8.8	0	0	92.8	2.5
MM 2B	9	48	0	0	0			14	35	0	0	3	3		0
3A	17	36	5	0	4			14	31	0	0	2	9		0
3B	8	15	0	0	0			21	55	0	4	11	4		0
Tot.	34	99	5	0	4	0	142	49	121	0	4	16	16	206	0	348
%	9.8	28.4	1.4	0	1.1	0	40.8	14.1	34.8	0	1.1	4.6	4.6	59.2	0.0
MiP 2A^c								0	13	6	0	0	0		0
3B^d								26	102	20	30	0	5		0
Tot.	0	0	0	0	0	0	0	26	115	26	30	0	5	202	0	202
%	0	0	0	0	0	0	0	12.9	56.9	12.9	14.9	0	2.5	100.0	0.0
FM 2A^e								5	66	0	8	0	0		0
3B	0	0	0	15	4		19	8	271	0	38	47	0		50
Tot.	0	0	0	15	4	0	19	13	337	0	46	47	0	443	50	512
%	0	0	0	2.9	0.8	0	3.7	2.5	65.8	0	9.0	9.2	0	86.5	9.8
All	34	114	5	15	8	0	176	114	815	26	108	63	21	1147	58	1381
%	2.5	8.3	0.4	1.1	0.6	0	12.7	8.3	59.0	1.9	7.8	4.6	1.5	83.1	4.2

^{a, b, c, d, e} n = 0 for each subcategory within TMSM/AMAP.

^f n = 0 within this subcategory across all books

Table 6 shows the prevalence of the three CMs in the form of (1) the number of responses linked to the respective CMs and subcategories, and (2) the proportion of these responses in the individual books, giving an indication of a preferred sequencing in fraction content and an indication of the type of fraction knowledge and what CMs for fraction are given precedence in each book series.

Results

As seen in Table 5, AOC is introduced across all books on average near the middle of the book series where the book MM does so somewhat later compared to the other books. On the other hand, TMSM/AMAP in this series are used with number lines a few pages before AOC. This is thus the only textbook that introduces fractions using TMSM/AMAP and the number line. Two other series (MDT and FM) introduce TMSM/AMAP around a third into the book, and later. One series of books (MiP) does not introduce TMSM/AMAP at all, focusing only on AOC.

Table 6 presents the number of responses and their respective proportions in the books. Books containing no responses concerning AOC and TMSM/AMAP have been excluded from the table.

All responses (n = 1381), which concern the three CMs, are found in the books for years 2 and 3. Of these responses, 83.1% (n = 1147) are responses linked to AOC and 12.7% (n = 176) constitute responses linked to TMSM/AMAP. A small proportion of tasks require responses using all three CMs, for instance by connecting a circle partitioned into thirds with “${\displaystyle \frac{1}{3}}$” marked, with a place on a number line (4.2%, n = 58).

Of the total number of responses in the MDT book series (n = 319), 4.7% are categorised as TMSM/AMAP, 92.8% as AOC and 2.5% as responses that use all three CMs. The book series FM display a similar pattern, where 3.7% of the total number of responses (n = 512) concern TMSM/AMAP, 86.5% concern AOC and 9.8% of responses use all three CMs. In this context, MiP stands out, where all responses (100%, n = 202) concern AOC. In contrast, a more even distribution (in this context) is found, of the total number of responses in MM (n = 348), between TMSM/AMAP (40.8%) and AOC (59.2%).

Overall, the AOC metaphor is more common in the books. One consequence of this general pattern is the presence of tasks involving colouring, labelling or marking one or more parts of a geometric shape as fractions (n = 815). These tasks account for 59% of all responses. Partitioning responses amount to 8.3% (n = 114), whilst comparing responses (and finding equivalence) amount to 4.6% (n = 63). There are very few responses requiring unitising actions (n = 21, 1.5%) and 7.8% of all AOC responses (n = 108) concern arithmetical operations.

Partitioning responses (n = 34) with the number line amount to 1.5%. Placing or labelling fractions on a number line between 0 and 1 is on the whole a rare exercise (n = 114), with 8.3% of all responses and 0.4% concerning mixed numbers on number lines (n = 5). Tasks requiring comparison and equivalence responses are also rare (n = 8, 0.6%) as well as arithmetical operations on the number line (n = 15, 1.1%). No tasks are found where unitising actions with the number line is required, for instance re-partitioning a number line, creating new subdivisions or finding the unit.

Although there are major differences between the book series – where MM has several tasks requiring answers related to TMSM/AMAP, MDT and FM have a few and MiP none – the general pattern is that these two CMs are less common. A common exercise in FM is to name/place/colour fractions (n = 337, 65.8%), whilst it is less common (but still significant) in MiP (n = 115, 56.9%) and in MM (n = 121, 34.8%).

Discussion

The following discussion will focus on the prevalence of the three CMs, possible issues related to this prevalence, and how these results can be understood in light of the process-object dimension of concept formation.

Conceptual Metaphors

The most common introduction of fractions in these textbooks is done through AOC. This result is in line with international research which shows that part-whole is a very common sub-construct in textbooks for years 2 and 3 (e.g., Alajmi, 2012). These findings also confirm assumptions about the introduction of fractions in Swedish schools (e.g., Ahl & Helenius, 2018). Although there are differences between the book series, it is clear that few tasks in general concern partitioning of area representations, the comparison of fractions using these representations, and unitising actions. The importance of these aspects of fractions has been highlighted in previous research (Tunç-Pekkan, 2015), whilst scholars also point to the risks of an over-reliance on the continuous part-whole model (Zhang et al., 2015). This result implies such an over-reliance.

The findings show that students working with two of these book series will encounter very few tasks concerning TMSM/AMAP and in one book series will not encounter these CMs at all. Number lines seldom go beyond 1, and students are seldom given the opportunity to compare fractions on a number line.

The CMs TMSM/AMAP in one book series (MM) are introduced before AOC, as early as in year 2, and in this series students will participate in exercises concerning the partitioning and creation of number lines, placing simple and complex fractions on number lines, and will meet number lines going past 1. This result is somewhat unusual compared to the other books, as well as from an international perspective where studies show that TMSM/AMAP are often introduced in year 3 or 4 textbooks (e.g., Watanabe et al., 2017). The CMs TMSM/AMAP with associated number lines are relevant content in relation to research that indicates the importance of understanding fractions as measure and as numbers with a location on the number line (Lamon, 2012). Whilst early introduction is possible, these types of tasks can be difficult for young students, (see Dougherty & Venenciano, 2007; Simon et al., 2018). Although the proposed order of introduction of CMs for fractions need not in itself be a problem – especially regarding instruction beyond the textbook – it is important from a conceptual change perspective (Vamvakoussi & Vosniadou, 2004) to reflect on which conflicts can occur in students’ learning when one CM is given priority over others.

Hence, there is a great inequality concerning TMSM/AMAP between the book series. There is also a tendency to over-rely on AOC with an abundance of labelling exercises (59%). As previous research and reports concerning the role of textbooks in Swedish classrooms show (Jablonka & Johansson, 2010), an overall indication is that these books will potentially create different opportunities to learn. This, in turn, suggests that teachers have a responsibility to actively reflect on the presence of these CMs for fractions in instruction and consider whether the chosen textbook fulfils the desired objectives.

Process-Object

Based on the assumption that process conception precedes object conception, it is noteworthy that only a small number of tasks focus on an actual process, for instance, the partitioning of objects (8.3%) or the creation of number lines (2.5%). By far the most common task is the labelling/colouring/selecting of a partitioned geometric shape (59%). However, compared to the task framework, these tasks correspond to the interiorisation stage of concept formation, and can thus be seen as appropriate, whilst it could be argued that these are tasks concerning processes on lower-level objects, that is, simply naming or pointing out fractions.

Few tasks can be connected to the condensation stage, for instance, the placing of fractions on number lines (8.3%), and almost no tasks are found on the reification stage, for instance, unitising objects (1.5%). These discrepancies may be due to the framework and textbooks’ different target groups of students (year 5 vis-à-vis years 1–3), whilst the textbooks’ perceived lack of a sufficient variety of tasks with respect to reification could be seen as a shortcoming.

If the concept of fractions originates in experiences of partitioning objects or segments (Lakoff & Núñez, 2000), a procedural activity in which different procedures produce different sized fractions (Pantziara & Philippou, 2012), and the operational conception precedes the structural conception (Sfard, 1991), then the fact that partitioning and unitising tasks are rare, is notable. Textbook creators have, to a limited extent, taken into account the embodied processes and CMs that form the basis of the concept of fractions.

With regard to the inherent duality of the fraction concept, labelling tasks are a reasonable inclusion in books for young students. It is nevertheless fair to question the overabundance of these tasks, as the process dimension is forgotten.

Limitations and Conclusions

The above findings are understandable in the context of the Swedish syllabus for years 1–3, which explicitly states that instruction should treat fractions as part-whole. Textbook authors should, however, be aware of the pitfall of over-reliance on AOC and part-whole area representations, whilst paying attention to the various aspects needed to fully use this sub-construct. The general absence of TMSM/AMAP and number lines in the textbooks is also understandable, whilst the number line’s importance for a full understanding of fractions should not be forgotten in instruction.

This study has not considered accompanying teacher guides or other teaching materials useful for instruction. Nor is anything said about teacher-constructed exercises or how other aspects of fractions fit into an overall picture of fraction instruction made possible by these books. Nor can we know if teachers use the book from beginning to end and allow students to work their way through the book sequentially. However, based on previous research concerning textbook hegemony in Sweden (Jablonka & Johansson, 2010), it seems reasonable to assume that these textbooks will produce different instructional practices.

The differences between textbooks need to be made visible for the benefit of said practice, and this study reminds us of the importance of opportunities and adequate systems to be in place for Swedish schools and teachers to be making relevant and conscious decisions in their choice of textbooks. This study also shows, in a partly novel way, how to use CMT and the metaphor concept as framed within this theory to demonstrate the concept’s relevance for education research and textbook analysis.

Disclosure Statement

No potential conflict of interest was reported by the author(s).