Textbooks are a common teaching tool widely used in children’s mathematical education. Comparative studies of textbooks have focused on different aspects, such as content, mathematical symbols and so on. However, a multimodal approach to textbook research—that is, studying how writing, images, mathematical symbols, etc. interact—is sparse. This study analyses 40 exercises from 17 Swedish Year 1 (children 7–8 years) textbooks using a multimodal approach with a focus on subtraction as an arithmetic operation. The aim was to describe and analyse how subtraction in Swedish Year 1 mathematics textbooks can be understood using a multimodal approach. The results show that it is sometimes possible to solve an exercise without focusing on the mathematical content that the exercise is designed to offer. Writing, images, mathematical symbols, speech and moving images are used differently within the same textbook and between textbooks. The results also show that there are considerable similarities between the exercises in printed and digital textbooks, with some exceptions. The examples in the study indicate that three different approaches are needed when working with these exercises, which implies great complexity in children’s meaning making in their work with mathematics textbooks. This could negatively impact children’s access to beneficial learning situations. Therefore, this study could contribute to a larger awareness of the complexity in question, which, by extension, may contribute to the development of beneficial learning situations in mathematics education, especially regarding subtraction.

In today’s society, we constantly encounter information in the form of different sign systems, whether in books or on television, the Internet, billboards or smart phones. Reading not only entails interpreting written text, but also, for example, images and sound (

Over 75 percent of children in compulsory education worldwide are taught mathematics with textbooks. In Sweden, that number increases to over 90 percent (

The aim of this study is to describe and analyse how subtraction in Swedish Year 1 (children 7–8 years) mathematics textbooks can be understood from a multimodal approach. This opens up the possibility of studying each resource in the textbook at the same time, but also these resources in combination. The research questions concern how subtraction is presented in printed and digital textbooks from different modes: ‘Which modes are used?’, ‘How do different modes interact with each other?’ and ‘What meanings are the exercises designed to offer?’

Textbooks can be understood as teaching tools intended to support children’s meaning making and, by extension, their learning. Using textbooks can mean more or less work for the learner, depending on the textbooks’ design regarding the use of different modes (

Existing research on mathematics textbooks often focuses on textbook analysis or textbook comparisons (

Thus, multimodal research on textbooks is scant, and multimodal studies focusing on subtraction as an arithmetic operation have not been found. Dowling (e.g.,

O’Keeffe and O’Donoghue (

An Indonesian study (

Mathematics textbooks ‘are tools with constraints and weaknesses’, Johansson concludes in her study of three Swedish teachers’ use of textbooks in Years 8 and 9 (

This study derives from a multimodal design for learning approach (

This study focuses on mathematical textbooks as multimodal teaching tools and their potential for meaning making. A multimodal approach could broaden the scope of how mathematics can be represented, according to Björklund Boistrup (

Modes carry potential for meaning making, and meaning making occurs in communication (e.g.,

Communication can occur between individuals or between an individual and an object, such as a textbook. When an individual encounters and interprets a textbook, meaning is made through representations of some kind—for instance, by showing with the body or by writing mathematical symbols. For example, a child reads a task in her textbook, counts on her fingers and writes down the answer ‘5’. In this case, meaning is made using gestures and mathematical symbols. The individual is represented here as a child but could also be the teacher, as the encounter between the teacher and the textbook also constitutes meaning making. However, this aspect will not be included in this study. Morgan (

To understand textbooks as teaching tools for meaning making from a theoretical perspective, Bezemer and Kress (

It is important to emphasise that this study does not focus on all meaning potentials. Textbooks can be considered teaching tools designed to offer a specific potential for meaning making to solve exercises in such a way that the content the exercise was designed to offer is discovered. This potential for meaning making is this study’s focus. The meaning individuals actually discover in their encounter with textbooks will therefore not be studied, but will be the subject of a forthcoming study.

In the literature, several researchers have described different subtraction situations. For instance, Carpenter and Moser (

Concerning subtraction in Swedish Year 1 textbooks, most exercises do not address a specific subtraction situation (Norberg & Boström, submitted). The study shows that if this is done,

With a multimodal approach comes an awareness of various modes, which guides the method of analysis. The method of analysis aims to make modes visible, as well as drawing attention to the potential for meaning making when working with mathematics textbooks. It must also illuminate the mathematical content being studied: subtraction. The analysis proceeds from Danielsson and Selander’s (

Exercises from various Swedish Year 1 mathematics textbooks on the Swedish market were chosen based on the following selection criteria. First, publishers were first chosen from an Internet search using the Swedish keywords for ‘textbooks’, ‘publishers’ and ‘elementary school’. Second, the Swedish Textbook Authors’ Association was contacted, and publishers who publish textbooks were identified. Third, this information was then checked with a reference group from a primary school, leading to the identification of six publishers. All textbooks produced from 2011–2017 were chosen. The year 2011 was chosen because it was when the most recent Swedish curriculum was published. Five textbook series had complete digital textbooks, and four of these also had a printed textbook, while one was available only in a digital version formatted for tablets. Altogether, 17 textbook series were studied: 12 in printed format, four in printed and computer-based formats and one in tablet format.

First, all pages containing subtraction in all textbooks (approximately 1,700 pages) were studied to gain an understanding of how subtraction is presented in Swedish Year 1 textbooks overall. The results from this survey are presented in a quantitative study (Norberg & Boström, submitted). Second, two to four subtraction exercises from each textbook were chosen for deeper analysis, totalling 40 exercises. The criteria for inclusion were that the exercises should derive from a subtraction situation, show breadth in the way that subtraction was presented and derive from the quantitative study’s results. The breadth was addressed according to subtraction situation and the way modes were used in the exercise. Some exercises are presented in the results section below with the publishers’ permission. These exercises were chosen to give the reader examples of the variation of exercises that exist. The first exercise is an example that can be a bit problematic. The second exercise is an example that works well, and the third exercise is an example that shows the printed and digital versions of the same exercise. Together, these three examples were also chosen to illustrate both two subtraction situations.

To understand how textbooks are structured, Danielsson and Selander’s (

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Each question was asked, and the answers were summarised. Information found in each mode and the potential for meaning making according to the subtraction content that the exercise was designed to offer was sought. If the text, for instance, said ‘Take away’ or ‘Compare’ or the image showed apples and apple cores, then it was possible to tell what kind of subtraction situation that the exercise was designed to offer. With these questions, a focus on different aspects of multimodality from a mathematics educational perspective was possible. This created an instrument for understanding the textbooks in a new way.

The aim of this study was to describe and analyse how subtraction in Swedish Year 1 mathematics textbooks can be understood using a multimodal approach. In this section, three examples of analysed exercises will be presented to illustrate the contribution of a multimodal approach to mathematics textbooks. The presentation structure follows the analysis questions above. After that, some general results will be given based on the analysis of 40 exercises taken from 17 textbook series to broaden the scope of the potential for meaning making in subtraction in Swedish Year 1 textbooks.

The first example (see Figure

Ducks, subtraction as

To understand this page as a multimodal teaching tool, it will be examined from the top to the bottom. The first, question was

The next section on this page starts with a small picture of two children talking to each other. It is explained in the beginning of the book that this is a symbol for working in pairs, so the image is used as an instruction. The writing reads, ‘Tell about the images. Write in maths’ language’. The images show three situations involving ducks grouped in different ways, as well as some grain, two slices of bread and breadcrumbs. Mathematical symbols are used to write the answers. In this section, there is congruence between the modes. To perform the calculation, the child must pay attention to the writing and images in this section. At the bottom of the page, the writing says, ‘Subtraction 0 to 5’, and the purpose of this text is to tell what chapter the page belongs to. Therefore, to summarise this page,

The third question was

The fourth and fifth questions were:

This exercise demonstrates that there is sometimes incongruence between the modes in the tasks and that more information may be included than is necessary.

In this example (see Figure

Dots and stars, subtraction as

To solve this exercise, the child must interpret the image as a resource for calculation. The dots and stars are supposed to be used as counters. The child should cross out the right number, then count the ones that remain and write the correct mathematical symbol on the empty line. Compared with the previous example, this exercise is plainer, as the whole page is composed of the same kind of exercise, and the images do not include any decorative elements. The child must focus on the mathematical symbols, except in the first task, where the counters have already been crossed out. The image supports the mathematical symbols and can be used as a resource for calculation, but it is nevertheless possible to solve the tasks without using the images, and the child should, if using the images, begin reading in the mathematical symbols mode. The approach needed to solve this exercise involves understanding the image as a resource for calculation, except in the first task, where the image is used as an instruction showing a sequence of events. The child is supposed to use the images to perform the calculation. This differs from the former example, where the child instead should understand the tasks as episodes that should be interpreted. This indicates a clearly different potential for meaning making, as the images in this example are designed to be used to perform a calculation that has not been done, whereas the former example requires the child to interpret a calculation that has already been done. These examples show that different approaches are needed to solve subtraction exercises encountered in mathematics textbooks.

Regarding the subtraction situation that the exercise is designed to offer, the writing and image in the first task inform the reader that this is a

This exercise provides an example of the fact that a certain order exists in how the exercise should be read to discover the mathematical content of the exercise.

The examples shown here derive from the printed (see Figure

Polly and Milton 1, subtraction as

Polly and Milton 2, subtraction as

To solve the exercise, mathematical symbols are used to write the answer. In both the printed and digital versions, writing, images and mathematical symbols are used for explanation and instruction. The images mode and the mathematical symbols mode are also used as resources for solving the task. There is congruence between the modes in the printed textbook, but the speech mode in the form of text to speech (TTS) creates incongruence in the digital version. The option to have the exercise read aloud in the digital version could be useful for the child, as this would provide support for interpreting the exercise. However, to do so, the TTS needs to be congruent with the other modes. In this exercise, the speech and the writing are incongruent because the voice of the TTS sounds monotonic, some writing is spoken incorrectly, and some is not read aloud at all. In TTS, the writing in the blue box reads, ‘Subtraction, Compare. We use subtraction left parenthesis minus right parenthesis when we compare and look at the difference. Polly has 5 caramels. Picture. Milton has 3 caramels. Picture. How many more caramels has Polly got? Picture’. The parentheses are read aloud, which could interfere with the child’s understanding of the text. At the end of the explanation, when different mathematical terms are explained, the TTS only says, ‘Picture’, which means that important information is left out. This indicates that the speech mode, which could be helpful when interpreting the task, does not work in this example. In fact, the speech mode makes it harder to discover the potential for meaning making needed to solve the exercise as designed.

As this page introduces subtraction as a

These pages are designed to offer subtraction as

Among other things, this exercise offers an example of the fact that the printed and the digital textbooks offer similar exercises and that improvements can occasionally be made in how modes are used in mathematics textbooks.

The modes used in the printed textbooks are writing, images and mathematics symbols, and in the digital textbooks, speech and sometimes moving images. The similarities between the exercises that are available in printed and digital formats are large, and the exercises in the digital textbooks look a lot like the exercises in the printed textbook, only on a screen. However, there are differences due to the possibility, for example, of having the writing read aloud in the digital textbooks, and sometimes short videos explain the concepts. The results show that improvements to these modes are sometimes needed. One type of digital textbook differs from the screen format, however: a tablet textbook. In this case, the layout benefits from a digital format, such that the exercises often show some kind of event in moving images, and the child can sometimes move objects, implying other postentials for meaning making than in other digital textbooks.

The modes are used differently and are sometimes congruent, sometimes not. To discover the mathematical content, different modes and different numbers of modes are used. The images mode is often involved, and sometimes more information is included than necessary. The results show that different modes can sometimes be considered unequally important for solving tasks and that, occasionally, the reading direction between different modes is important when solving exercises. The results also show that the approach needed to solve the exercise differs not only between exercises, but also within the same exercise, between textbooks and within the same textbook. In total, three different approaches are necessary to solve the exercises: an episode that should be interpreted, a resource for calculation and a guide box. This implies that the potential for children’s meaning making in their work with textbooks is complex, and an awareness thereof is necessary.

This study has focused on how subtraction in primary school textbooks can be understood using a multimodal approach as a way of researching potential for meaning making. Different modes contribute different potentials for meaning making. For example, in ‘Dots and stars’, the writing mode conveys information about how to solve the task. The mathematical symbols mode is used to understand what calculation to perform, and the images mode is used as a resource for calculation. These modes complement each other. This is in line with the results of Nugroho’s (

The ‘Ducks’ example is an exercise in which more information is given than is needed. This could mislead children, as the images mode shows ducks walking and standing with their beaks pointing in different directions and with grain, slices of bread and breadcrumbs that could hamper the interpretation. This is supported by Sutherland, Winter and Harris (2008) and could be compared to Nugroho’s claim that images concretise mathematical concepts for children. It is possible that the notion that images make it easier to interpret mathematics tasks may however constitute a simplification of the impact of images on children’s interpretations.

The results of this study show that an incongruence sometimes appears between different modes, which could be confusing for the child. An example of this is given in the ‘Polly and Milton’ example between the speech and writing modes. If the potential for meaning making to solve the task with the images mode points in one direction, and the writing mode points in another, the task will be difficult for the child. Unnecessary energy could also be exerted in understanding what to do with the exercise instead of practicing, in this case, subtraction. This can be seen in the ‘Ducks’ example, in which there is incongruence in the writing mode in the use of ‘subtraction’ and ‘minus’. This incongruence is especially important, as this is the concept that is to be taught, which could mislead the child. This result is supported by Dyrvold (

The results demonstrate that there could be a focus on how the exercise should be read in order to solve the tasks. This means that the child might need to begin reading in one particular mode. For instance, in the ‘Dots and Stars’ example, the child should proceed from the mathematical symbols mode to solve the tasks, whereas in the ‘Ducks’ example, the images mode is the starting point. In other examples, such as ‘Polly and Milton’, the child can choose between the images and mathematical symbols mode. To solve the task as designed, the child needs to know when to read in what way. This work could be energy-consuming for the child, and this supports the concept that mathematical textbooks require more work from the learner (

Regarding which mode or modes carry the mathematical content focused on in the exercise, the results show that different modes and different numbers of modes are used, and the images mode is often involved in the mathematical content design. The same modes could convey information for both solving the task and discovering the subtraction situation, but that might not be the case. A conclusion to be drawn from this is that the child might solve the tasks without noticing the subtraction situation that is intended to be taught. The child might also need help to identify which modes provide information for solving the task and which modes convey the mathematical content.

The examples in this study indicate three approaches needed to solve the exercises: as an episode that should be interpreted, as a resource for calculation, and as a guide box. In turn, examples of these are shown in the lower part of the ‘Ducks’ example, the ‘Dots and stars’ example and in the upper part of the ‘Polly and Milton’ example. To solve the tasks, the child must be aware that there are different ways of interpreting the tasks. When comparing two of the exercises, for example, in the ‘Dots and stars’ example, the images are designed to be used for calculation. In the lower part of the ‘Ducks’ example, the images are designed to be used as episodes that should be interpreted. Thus, then the intended way to interpret the former exercise would not be applicable to the latter. Another example of this is given in the upper part of the ‘Ducks’ example. This part is designed to be interpreted as a guide box. A child who does not read this as it is designed could lose the opportunity to identify a bearer of important information and pay little or no attention to it, as there are no tasks to solve on that part of the page. Modes are also used differently within the same textbook and between textbooks. One conclusion to be drawn from this is that different potentials are needed when working with subtraction exercises in primary school Year 1.

The results show considerable similarities between digital and printed exercises from textbooks available in printed and digital formats. What differs most between the digital and printed exercises is the possibility of having exercises read aloud and the fact that the former is in a digital format. According to how different modes are used, the results give examples of inadequacies in the use of speech, as in the ‘Polly and Milton’ example. This can imply that digital textbooks could be developed with better TTS. Another development opportunity related to digital textbooks is the possibility of showing processes—for instance, in a subtraction situation in which the moving images mode can be used to illustrate the situation. This would entail the child receiving support in interpreting the subtraction situation instead of trying to figure the process out by reading still images.

In summary, this study shows that writing, images and mathematical symbols are used differently in different exercises both within a textbook and between textbooks. The results also show that it is sometimes possible to solve an exercise without focusing on the mathematical content that the exercise is designed to offer. Three approaches are needed when working with the analysed exercises in this study: an episode that should be interpreted, a resource for calculation, and a guide box. This implies great complexity in the potential for meaning making in children’s work with mathematics textbooks. This mode issue may deny children access to beneficial learning situations unless teachers, publishers and textbook authors are made more aware of how different modes and meaning potentials contribute differently to children’s possible meaning making in subtraction exercises. Therefore, studies like this are of importance for the development of beneficial learning situations in mathematics education, as they emphasise awareness of the complexity of meaning making when working with mathematics textbooks. Teachers could invite children to participate in discussions that position the textbooks as multimodal texts and help them focus on those aspects, thereby facilitating children’s meaning making and, by extension, learning.

The author has no competing interests to declare.