Introduction

Mathematics curricula worldwide emphasise mathematical communication as a central competence in mathematics teaching and learning, often stating that students should develop their ability to communicate orally and in writing. While oral mathematical communication in a school setting has received much attention recently, written mathematical communication has been backgrounded. This article attempts to review and discuss what is known about mathematical writing in a learning context.

Given that students’ mathematical writing has been used throughout history to assess several mathematical abilities, the question of how students should be taught to write—to learn to write rather than write to learn—should attract significant interest among researchers in mathematics education (Barwell, 2018; Morgan, 2001b). So far, however, this has not been the case. In mathematics, students are often asked to solve a mathematical problem and hand in a written account of their process, where they ‘show their thinking’ without separating the two processes of (i) solving the problem and (ii) writing about it. Morgan (2001b) has argued that mathematical writing is important enough to merit an interest beyond its indiscernible part of the problem-solving process, thus suggesting that the ability to communicate through writing in mathematics is an ability that can be separated from other mathematical abilities. Existing research on students’ mathematical writing competence and strategies for developing this is limited. Teachers are often left with a limited toolbox to assess and help students improve the quality of their writing (Colonnese, 2020; Namkung et al., 2020; Teledahl, 2015).

One possible reason for mathematical writing being an under-researched issue, especially when it comes to assessment and teaching, is the tension that can be found between conventional, formal forms of mathematical writing and students’ idiosyncratic and personal ways of making meaning in writing (Sfard et al., 1998; Smith, 2003; Steenrod et al., 1973). Barwell (2005) argues that this tension can be traced back to two different conceptualisations of writing: the formal model and the discursive model. In the formal model, mathematical writing is regulated by a stable set of rules. Students are expected to learn the rules, and their command is measured in relation to the number of mistakes they make. In contrast, in the discursive model, mathematical writing is part of a social and discursive practice. Social structures shape meaning, and rather than being fixed, the rules are negotiated and determined in the social context of, for example, the classroom. In the discursive model, acceptable or good writing varies with the situation (Barwell, 2018).

It should be noted that the multi-semiotic nature of mathematics, where different representations, such as numbers, symbols, and images, as well as natural language, are used to make meaning (O'Halloran, 2005; Steenrod et al., 1973), contributes to the tension between the two ways writing can be viewed. The tension also parallels the philosophical question of whether mathematics can be seen as an autonomous system, independent of humanity, or a human activity (Morgan, 2001a). A researcher who has suggested a way to approach these tensions in the case of mathematical proof is Stylianides (2007). From an investigation of third graders’ work with proofs, Stylianides decomposes the conventional requirements of mathematical proof and transforms them to function in a school context that includes young students. He shows that it is reasonable for third graders to produce understandable proofs to their peers while concurrently complying with universal and general requirements for mathematical proofs.

Suppose we move from proofs to a broader view of mathematical writing. In that case, it can be argued that between the two conceptualisations and the philosophical positions, it is difficult to define ‘good mathematical writing’ in a way that makes it possible to assess and teach. To teach students how to write mathematical texts that can be read and understood by others and outside of their immediate context, teachers need to attend to formal rules and conventions and promote the creative use of various representations. For this, teachers must understand how quality in students’ writing can be defined, described, and measured. In a school situation, the different types of mathematical writing include exploring and discovering, explaining and describing, critiquing and constructing arguments, and creating and extending (Casa et al., 2016) or simply keeping notes and making calculations while doing mathematics. In some situations, students write for themselves while they, in other situations, are addressing a potential reader. The reader of most types of writing in the mathematics classroom is the teacher, who may foreground communicative aspects of the writing to understand what the students have done. Other quality aspects of mathematical writing may concern the avoidance of redundancy, the use of symbolic language, and the adherence to formalistic traditions. Addressing the quality of students’ writing in mathematics is also a matter of differentiating between issues of syntax (structure and grammatical rules) and semantics (the meaning of the various text elements). Given the limited research on students’ mathematical writing, it is reasonable to assume that quality in students’ mathematical writing is neither well-defined nor easy to measure, which is why we address these issues.

This literature review aims to identify aspects of quality that have been or could be addressed in teaching and assessing students’ mathematical writing. Towards this aim, we investigate current research on mathematical writing in search of answers to the following questions:

  1. 1.

    What types of mathematical writing are researched?

  2. 2.

    How do researchers define and measure quality in mathematical writing?

Method

The current study focuses on English-language, peer-reviewed empirical research articles concerning guidelines or frameworks for writing in mathematics education. To structure our review study, we followed the Preferred Reporting Items for Systematic Reviews and Meta-Analysis (PRISMA) guidelines (Moher et al., 2009). We used the software Covidence to manage our data.

Data selection

A systematic literature search was performed on May 24, 2021, in ERIC (EBSCO), searching abstracts using the following manually composed search string: [AB writing AND (mathematical OR mathematics) AND (pupil* OR student* OR children*)]. The selection of articles followed the four main steps of PRISMA: (1) identification, (2) screening, (3) eligibility, and (4) inclusion (Moher et al., 2009). With no publication date restrictions, our identification phase embraced studies at all levels of mathematics teaching. Excluding editorials, book chapters, conference proceedings, and articles published in teacher journals, as these are not always peer-reviewed, we identified 806 peer-reviewed articles published in English.

A screening process was conducted in two steps. First, a title and abstract screening was done based on the inclusion and exclusion criteria in Fig. 1 (IC1–3.EC1, EC2). This step resulted in a data set of 141 articles for which a full-text screening was conducted. We refined our exclusion criteria so that research studies were excluded if they did not contain criteria for analysing or assessing mathematical writing, for example, frameworks, guidelines, or progression (Fig. 1, EC4). Articles were also excluded if there was no full text available and if it was not about mathematical writing (Fig. 1, EC5–6). After this screening, 49 studies remained. Throughout the screening process, each study was screened by two researchers separately, and disagreements regarding inclusion or exclusion were discussed. Data extraction included general information about the author(s), title, purpose statement(s), the country where the study was conducted, research question(s), methods, target group, and results. A final quality appraisal was conducted to establish eligibility, resulting in the exclusion of five articles. This resulted in a total of 44 articles (Fig. 1).

Fig. 1
figure 1

Flow diagram of the selection process for the original search. Note that the new search performed in December 2023 was identical to the search described here and resulted in the inclusion of four more articles

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On December 21, 2023, a new search was conducted to cover the period between May 2021 and December 2023, resulting in an additional 87 articles. An identical screening process was performed for these articles, resulting in 4 more articles, which means that, in total, 48 articles were included in the final data analysis.

Process of analysis

The analysis of each article was conducted in six steps, as described below.

Step 1: Each article was read, and a short description was made of…

  1. i)

    …what kind of mathematical text the study concerned.

  2. ii)

    … what aspects of the text were analysed.

  3. iii)

    … what kind of analysis was conducted.

Step 2: Existing categories of types of writing (e.g. Casa et al., 2016) were tried but did not provide sufficient differentiation or coverage. Consequently, an inductive analysis was conducted. From the descriptions produced in step 1, a constant comparative method (Corbin & Strauss, 2008) with open coding was used to compare the different descriptions of kinds of texts. Through an iterative process of identifying similarities, five types of texts emerged, coded from A–E:

  1. A.

    Descriptive writing in problem-solving situationsFootnote 1

  2. B.

    Proof writing

  3. C.

    Reflective writing (journal writing/blog)

  4. D.

    Expository writing

  5. E.

    Mathematical writing in general (student written work in connection to diverse tasks).

Step 3: Using the codes from step 2, the first and second authors re-read and re-coded each article according to what type of mathematical text the study was about.

Step 4: The two researchers’ coding from step 3 was compared, and differences were discussed to reach a consensus. There were mainly two issues discussed. The first was an initial attempt to create subcategories for A because it was a much larger category than the others. These were difficult to distinguish, so all were coded as A. The second issue concerned how to code articles where the texts in question included explanations of problem solutions and, therefore, could be seen as either A or D. We decided to code them as A if they were about describing and explaining a solution to a problem and as D if the prompt explicitly explained a mathematical idea or concept (see Table 1). After that, the results of both processes aligned, indicating that the codes were reliable to answer research question 1.

Table 1 Articles coded in each category
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Step 5: In each category, all the articles were scrutinised for similarities and differences to pinpoint essential characteristics. A descriptive narrative incorporating all the articles was written for each category to answer research question 2. When identifying different aspects of the texts included in a category, these aspects were not considered mutually exclusive but were used to highlight characteristics within the specific type of text.

Step 6: In the final step, the quality measures in each article were identified by investigating which quality aspects were described as hierarchical or valued differently. To conclude the analysis, levels of quality were described in more general terms namely text structure, character of writing, and text elements.

Results

The analysis identified five categories of types of writing, which will be presented below. Table 1 displays the number of articles in each category. These categories answer our first research question: What types of mathematical writing are researched? The second question, How do researchers define and measure quality in mathematical writing, is answered separately for each category. The last part of the results describes ways the quality measures are valued at different levels that could indicate progression.

Category A: descriptive writing in problem-solving situations

Cohen, et al. (2015). Characteristics of second graders’ mathematical writing

Craig (2011). Categorisation and analysis of explanatory writing in mathematics

Hauk & Isom (2009). Fostering college students’ autonomy in written mathematical justification

Hughes et al. (2019b). Evaluating various undergraduate perspectives of elementary-level mathematical writing

Ikhsan et al. (2020). An analysis of mathematical communication skills of the students at Grade VII of a junior high school

Johnson et al. (1998). Students’ thinking and writing in the context of probability

King et al. (2016). Promoting student buy-in: Using writing to develop mathematical understanding

Kiuhara et al. (2020). Constructing written arguments to develop fraction knowledge

Kosko & Singh (2019). Children’s coordination of linguistic and numeric units in mathematical argumentative writing

Kosko & Zimmerman (2019). Emergence of argument in children’s mathematical writing

Lim & Pugalee (2004). Using journal writing to explore “they communicate to learn mathematics and they learn to communicate mathematically”

Morgan (2006). What does social semiotics have to offer mathematics education research?

Pugalee (2001). Writing, mathematics, and metacognition: Looking for connections through students’ work in mathematical problem solving

Pugalee (2004). A comparison of verbal and written descriptions of students’ problem solving processes

Santos & Semana (2015). Developing mathematics written communication through expository writing supported by assessment strategies

Steele (2005). Using writing to access students’ schemata knowledge for algebraic thinking

Taylor & McDonald (2007). Writing in groups as a tool for non-routine problem solving in first year university mathematics

Teledahl (2016). How young students communicate their mathematical problem solving in writing

Toker (2021). Making thoughts visible through formative feedback in a mathematical problem-solving process

The first category is the largest, with 19 articles investigating students’ written work when reporting on problem-solving activities and describing the solution. It is difficult to identify a common idea on what reporting on problem solving entails regarding what, how, why, and to whom students write when they write down solutions to their problem solving. The lack of a common idea is evident as the articles targeting problem-solving report different aspects of students’ writing.

The most frequently analysed aspect is students’ use of language. The term language, even when specified as mathematical language, is rarely defined but implicitly covers students’ use of words and numbers. The common denominator for the studies that target language and representations is a complete or partial focus on the text itself. Figure 2 shows two dimensions of what is analysed or brought up in guidelines or frameworks. The first concerns whether the text as a whole or a separate text element is considered, and the second dimension concerns linguistic features. Thus, the figure illustrates four different approaches to language found in these articles, each described and exemplified below. Some articles cover more than one approach.

Fig. 2
figure 2

Aspects of students’ writing

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When analysing students’ use of language, many studies focus on surface and syntax aspects, which are often possible to assess objectively. Examples of such objective aspects are linking words to show reasoning (Cohen et al.) and the total number of words used (Kiuhara et al.; Cohen et al.). Two articles (Teledahl; Santos & Semana) analysed students’ choice of representations, including images and symbols. Less objective but still surface aspects are, for example, level of formality (Ikhsan et al.; Cohen et al.; Santos & Semana), accurate use of mathematical vocabulary (King et al.; Lim & Pugalee; Taylor & McDonald), and level of sentence complexity or completeness (Hughes et al.(b); Santos Semana; Cohen et al.). Primarily, studies focusing on syntax aspects address the text as a whole or give few details about their analytical procedures. Five articles stand out, often in addition to general language features, explicitly targeting specific elements in students’ texts. King et al. analyse the mathematical steps used in the solution process, while Taylor and McDonald focus on the elements of aim, method, results, and conclusion. Toker looks at the three elements: drawing, writing explanatory sentences, and explaining the process. Other elements that have been the object of analysis are justifications (Santos & Semana) and problem statements (Hauk & Isom).

A different way to analyse language in students’ writing is to target semantic aspects, looking at underlying meanings, done in five articles. For example, Craig looks at the roles of textual categories, such as examples, questions, and discussion, to classify students’ texts in one of the three categories: recount, summary, or dialogue. Along the same line, Johnson differentiates between recording, summarising, generalising, and relating, and Kiuhara et al. between the rhetorical elements of restate, reason, counterclaim, and conclusion. Morgan, following Halliday, suggests using systemic functional analysis to identify three metafunctions—the ideational, the interpersonal, and the textual in students’ writing. As with articles that focus on syntax, some articles that target semantic aspects investigate specific elements in the text rather than the text as a whole. Kosko and Singh and Kosko and Zimmerman target students’ arguments and aim to identify different levels of detail. Teledahl investigates how and for what purpose students use different representations in problem solving.

In contrast to the articles described above that focus on the text itself, we identified five studies focusing on what the students’ texts indicate in the form of students’ behaviour or understanding of problem solving. Studies in this group investigate possible connections between what students write and what they understand or how they behave during problem solving (Pugalee; Steele; Taylor & McDonald; Hauk & Isom). Pugalee, for example, used students’ written descriptions of their problem-solving processes to identify metacognitive behaviours such as analysis of information and conditions, assessing problem difficulty, drawing diagrams, organising data into other formats, and evaluating decisions.

Studies in this category rarely explicitly attend to questions about teaching students how to write when reporting on problem solving. In most studies, the assessment of mathematical writing is implicitly seen as an assessment of problem-solving abilities or mathematical knowledge. Often, these two are intertwined in the same study.

Category B: proof writing

Ararfaj & Sangwin (2022). Investigating a potential format effect with two-column proofs

Ko & Knuth (2009). Undergraduate mathematics majors’ writing performance producing proofs and counterexamples about continuous functions

Lew & Mejía-Ramos (2019). Linguistic conventions of mathematical proof writing at the undergraduate level: mathematicians’ and students’ perspectives

Selden et al. (2018). Proof frameworks: A way to get started

Four articles report on studies about proof writing. In these texts, the writing is an attempt at writing ‘for someone else’ since the whole point of proof is to convince ‘everyone’ that something is true. The recipient is somebody and everybody, albeit only implicitly present. It is also the only category focusing more on ‘learning-to-write’ than ‘writing-to-learn’. Drawing on three of the articles by Selden et al., Lew and Mejia-Ramos, and Ko and Knuth, we identify the following four issues of mathematical proof writing to be of interest from a teaching and learning point of view and helpful in assessing the quality of proof writing:

  • Defining mathematical objects (variables).

  • Producing a logically coherent narrative.

  • Obeying the rules of academic language.

  • Attending to the context in relation to formality.

The fourth article explores using a two-column format for proof writing, which helps students increase the overall frequency of justifications and the number of non-explanatory justifications. The coding scheme differentiated between principle-based explanations, goal-driven explanations, noticing explanations, and non-explanatory paraphrasing.

Category C: reflective writing

Baxter et al. (2005). Writing in mathematics: An alternative form of communication for academically low-achieving students

Clarke et al. (1993). Probing the structure of mathematical writing

Freeman et al. (2016). How students communicate mathematical ideas: An examination of multimodal writing using digital technologies

Gearing & Hart (2019). The impact of adding written discourse to six year olds’ mathematics explanations within a problem-based learning unit

Kostos & Shin (2010). Using math journals to enhance second graders’ communication of mathematical thinking

Nie et al. (2007). Application of generalizability theory in the investigation of the quality of journal writing in mathematics

Liljedahl (2007). Persona-based journaling: Striving for authenticity in representing the problem-solving process

There are seven articles coded as reflective writing. Five of them report on mathematical journal writing at different levels of the school system (Baxter et al.; Clarke et al.; Gearing & Hart; Kostos & Shin; Nie et al.), and one on the use of a digital notebook and a mathematics blog (Freeman et al.). These six articles all describe the aim of ‘writing-to-learn’ more than ‘learning-to-write’. Using mathematical journals is thought to help students develop their mathematical proficiency and mathematical thinking by reflecting on the lesson. The quality of the writing is evaluated in terms of mathematical correctness and inclusion of mathematical activities such as recounting (knowledge), summarising, generalising, relating, reasoning, questioning, understanding, and elaborating. These articles also discuss the journals as a means of assessment, giving insights into students’ mathematical thinking and understanding, and as a way for students to communicate with the teacher. Only Freeman et al. mention writing to communicate with peers, but that was on a blog where the communication tended to be primarily non-mathematical. Sometimes (for example, Nie et al.), linguistic issues of form, such as spelling, neatness, or specific wording, were explicitly excluded from the analysis since the aim was not the writing itself but the development of mathematical abilities. The seventh article (Liljedahl) is different. The study aims to teach pre-service teachers to write in a particular way using a persona framework to improve their understanding of their problem-solving process.

Category D: expository writing

Arsenault et al. (2023). Mathematics-writing profiles for students with mathematics difficulty

Ganguli (1994). Writing to learn in mathematics: Enhancement of mathematical understanding

Guce (2017). Mathematical writing errors in expository writings of college mathematics students

Kline & Ishii (2008). Procedural explanations in mathematics writing: A framework for understanding college students’ effective communication practices

Latulippe & Latulippe (2014). Reduce, reuse, recycle: Resources and strategies for the use of writing projects in mathematics

Shield & Galbraith (1998). The analysis of student expository writing in mathematics

Stonewater (2002). The mathematics writer’s checklist: The development of a preliminary assessment tool for writing in mathematics

Van Dyke et al. (2015). Conceptual writing in college-level mathematics courses and its impact on performance and attitude

According to Shield and Galbraith (p 30), ‘journal writing normally invites students to reflect on their learning by expressing their thoughts […while] expository writing is intended to describe and explain’. Often, some writing prompt is used to elicit an explanation. There are eight articles in our data coded as expository writing. The writing tasks in these studies aim to explain a mathematical idea or concept comprehensibly and efficiently so that someone else can understand. Not to ‘show your thinking’ or ‘solve a problem’ but to define, explain, and discuss.

In three studies (Kline & Ishii; Latulippe & Latulippe; Shield & Galbraith), the prompt is to write a letter to a fictitious friend or client. What is valued in these texts, as well as in a study of college students’ writing by Ganguli, can be compiled into three main themes:

  • Elaboration/articulation, for example, explanations, justifications, and links to prior knowledge

  • Aspects of mathematics, such as types of mathematics used, representations, examples, and correctness

  • Language use, including appropriate mathematical notation and vocabulary, grammar, and structure (described by Latulippe & Latulippe as professional).

In addition, Kline and Ishii included engagement, implying that the writer should consider the reader when communicating. Arsenault et al. studied explanatory texts by students with mathematical difficulties, scoring aspects of the first theme (mathematics content, organisation, and clarity) and the third theme (vocabulary and grammar).

Two studies investigate university students (Stonewater; Van Dyke & Stalling), both basing their analysis on a framework developed by Stonewater. The categories in this framework coincide well with the three themes mentioned above but also include the ability to build a context, perhaps more applicable to essay writing than other types. One exciting result of Van Dyke and Stalling’s study is that students in their writing groups showed a negative attitude towards writing in mathematics, and many did not, according to the teachers, satisfactorily complete their writing assignments although they thought so themselves. It was difficult for these students to see the point of writing and feel engaged. In contrast to the letter prompt, these essays and conceptual writing tasks lacked a meaningful recipient, and communication was only a surface feature.

One study in this category stands out because the focus is on errors in mathematical writing (Guse). From a pre-defined list of nine common errors, the findings showed that two dominated the students’ writing: misuse of mathematical symbols and incorrect grammar. Both errors refer to syntactic aspects of communication—in mathematics’ symbolic language and the natural language, respectively. Misusing the equal sign is the most prominent example of the first error type. Note that incorrect notation was considered a different error type, the least occurring error. This result indicates that students often know which symbols to use but do not always use them correctly.

Category E: mathematical writing in general

Hebert & Powell (2016). Examining fourth-grade mathematics writing: Features of organization, mathematics vocabulary, and mathematical representations

Hughes et al. (2019a). Exploratory study of a self-regulation mathematical writing strategy: Proof-of-concept

Kesorn et al. (2020). Development of an assessment tool for mathematical reading, analytical thinking and mathematical writing

Liew et al. (2022). Children’s errors in written mathematics. Children, 14(5)

Nachowitz (2019). Intent and enactment: Writing in mathematics for conceptual understanding

Ntenza (2004). Teachers’ perceptions of the benefits of children writing in mathematics classrooms

Ntenza (2006). Investigating forms of children’s writing in Grade 7 mathematics classrooms

Powell & Hebert (2016). Influence of writing ability and computational skills on mathematics writing

Seo (2019). An investigation of how 7th grade and 8th grade students manipulate mathematical writing elements

Özgen et al. (2019). An investigation of eighth grade students’ skills in problem-posing

Several studies look at a broader range of students’ mathematical writing, including everything students write in a mathematics lesson or on a test, posing various types of questions. For example, Nachowitz examines American elementary students’ mathematical writing to see if they align with the intentions prescribed by research and common standards, grading the level of epistemic complexity. Similarly, Ntenza studies the mathematical writing produced in six South African classes, classifying the use of language on a range from direct (copying or transcribing) to creative. Although the two studies are from different continents and 15 years apart, both Nachowitz and Ntenza conclude that the writing produced fails to live up to contemporary expectations.

Two studies single out particular types of writing. Özgen et al. look at students’ problem-posing and develop a rubric to evaluate various problem-posing skills. Seo investigates short writing from ‘exit tickets’ to see how students use three mathematical elements (symbols, nominalisations, and images). Seo’s results show that all students conformed with the teacher’s expectations, implying that elements of mathematical writing that we want students to develop must be made an explicit part of instruction for students to know how to use them.

In some articles, mathematical writing is assessed without detailing the assessment tool because they have a different aim, for instance, investigating relationships between mathematical writing and other variables (Hughes et al.; Powell & Hebert) or measuring the validity and reliability of an assessment tool (Kesorn et al.). Liew et al. look at errors made in written mathematics, identifying the lack of correct terminology and misuse of mathematical symbols as the most common errors among above-average students in grade 6. Their findings are similar to those of Guce, described under category D. Hebert and Powell investigate genres, comparing students’ mathematical writing with other types of writing. Looking at both word problems and fraction tasks, they evaluate organisational features of the text, such as the use of paragraphs and transition words, and mathematical features, such as the use of mathematical vocabulary, symbols, and representations.

Different levels of quality

Most of the studies present some progression or at least different ways to evaluate the quality of students’ mathematical writing. When assessing levels of quality, the studies attend to language use and mathematical or organisational features, often in combination. Many of the criteria revolve around the text’s structure. Others deal with the character of the writing, for example, what type of writing the text displays. Some criteria focus on including various semiotic elements in the text, an element here being a form of representation or a text component with a particular function.

Structure of the text

One way of expressing progression or levels of quality is to focus on how students have structured or organised their text to create a comprehensive ‘story’. This progression can be either qualitative or concerned with the existence or not of certain features, for example, whether or not the student structures the text by orienting the reader, organising the facts or different steps, showing calculations, or verifying results (Pugalee, 2001; Stonewater, 2002). An alternative to describing progression as a question of deciding if someone is, for example, building a context or not is to describe a progression qualitatively. Nachowitz (2019) provides an example when he differentiates between ‘separated pieces of facts’, ‘partial explanations and well-organised facts’, and ‘well-organised explanations’. This type of qualitative measure is typical, and in many of the studies, differences in quality are described using terms such as clear and understandable. However, such assessments seem subjective, and there are few explicit descriptions of what makes a text clearer overall.

The character of the writing

There are descriptions of a qualitative progression that deals with the character of the writing. Johnson et al. (1998) and Baxter et al. (2005) differentiate between recording, summarising, generalising, and relating, where relating represents the most advanced writing method. Ntenza (2006) provides a description of the progression from direct copying to translating, summarising, and interpreting, and finally to creative writing, where the latter is the most valued. Craig (2011), using Waywood et al., suggests a progression that covers recount, summary, and dialogue. More advanced writing seems to entail more creative work on behalf of the students and the impression that such texts result from a more significant effort or higher awareness of the texts’ communicability. However, whether this awareness is explicitly taught or results from a natural diversity among the students who participated in the studies is unclear. This way of describing progression can be seen as connected to why students write, which is rarely reported in the studies.

Elements in the text

Many studies provide descriptions of progression tied to students’ inclusion of different elements. Elements are either segments or components in which students have used different semiotic resources or text components with specific functions. Some studies describe the use of, for example, mathematical terms as an element, and progression can be seen as a question of the level of accuracy or the frequency with which mathematical terms are used (Seo, 2019; Lew & Mejía-Ramos, 2019; Lim & Pugalee, 2004). Images or pictures along with graphs, diagrams, and mathematical symbols represent examples of semiotic resources that are either valued in context, rendering qualitative measures that are connected to how well they create meaning (Powell & Hebert, 2016; Stonewater, 2002) or valued objectively yielding quantitative measures connected to whether or not, or how often, the semiotic resources occur (Cohen et al., 2015; King et al., 2016). The most common elements attended to are text components that can be identified by their function, such as explanations, justifications, or arguments. As with semiotic resources, progression is described both qualitatively and quantitatively. While quantitative measures deal with the occurrence of, for example, explanations, qualitative measures are strongly associated with terms such as clear, well-explained, or understandable. As with qualitative measures regarding structure, there are few descriptions or analyses of what makes an explanation clear or understandable. One of the very few examples of such an analysis is Kosko and Zimmerman (2019), who, by using ideas from Halliday and Toulmin, respectively, describe what increases the quality of a mathematical argument. One step away from a discussion on ‘clear and understandable’ is quality measures that are more connected to students’ grasp of the mathematics, such as ‘using one’s own words’ (Ikhsan et al., 2020) and using particular/generalised or procedural/descriptive language (Shield & Galbraith, 1998).

Discussion

We have seen that mathematical writing as a research topic covers various types of texts. One-third of the articles in our data set were concerned with problem solving and the ability to report on solutions to mathematical problems. Our initial attempt to identify subcategories in this group failed because it was often unclear what role the mathematical writing played. The only evident subcategory in which the role of writing was prominent was one article about problem posing. In the other 18 articles, different aspects of problem solving and reporting on solutions were intertwined or unidentified, for example, mixing up the activity (i) of writing while solving the problem, (ii) of writing to explain the thinking while solving the problem (the steps taken), (iii) of explaining the solution, and (iv) of justifying strategies chosen. In most studies, the mathematical writing asked of the students served a seemingly unclear purpose from the student’s point of view. The underlying assumption seems to be that students should report on solutions as a way of ‘writing-to-learn’ while teachers read these to assess mathematical knowledge. The quality aspects mentioned often emphasised syntax issues connected to both natural language and the symbolic language of mathematics. Little was said about the teaching of mathematical writing or about ‘learning to write’ when communicating a solution to a mathematical problem. Such a lack of focus on research-based basic steps in teaching mathematical writing leaves teachers with a limited toolbox and few opportunities to focus exclusively on writing as a learning object, separate from other objects such as problem-solving strategies or mathematical reasoning.

In the research on expository writing presented above, the communicative aspect of the text is more prominent than in other types of writing. Results indicate that for this type of writing to feel meaningful, students need a recipient, for example, a friend. In a teaching situation, this idea could be exploited further, having students communicate with peers in other classes or other schools to create authentic communication, where there is a realistic need for someone to understand what is written. For these situations to be helpful in terms of students learning to write, however, it could be argued that similar to teaching students how to write reports on problem solving, teaching them to explain something to someone requires them to imagine what their ‘target reader’ knows and what they need to be told. As was evident for most categories, there is minimal discussion on audiences or target readers for most of what students write. When the teacher is the target reader, many of the aspects that are naturally a part of communicating with others, such as clarity or efficiency, are unnecessary because the teacher can fill in many of the ‘blanks’ with their contextual knowledge as well as their knowledge of the student and her ability. Neither redundancy nor insufficient information becomes an issue when the context provides all the details that might be missing or when the teacher does not have to separate necessary from unnecessary information.

When the teacher is the target reader for students’ writing, students must also consider what is expected of them. Given that our review shows very little agreement on standards for mathematical writing beyond the correctness of the mathematical notation, it can be assumed that students’ choices depend very much on what they believe their teacher expects. Expectations can concern issues of syntax or semantics, issues of choice of representations, levels of formality, or even the total number of words students use. Combined with students’ writing often being seen as a means to assess their mathematical knowledge, the difference in expectations from one classroom to the next creates a complex situation in which learning to write in mathematics is challenging. It seems as if the only time standards for mathematical writing are clear and uncontested is when it is reduced to its symbolic elements, the ones used to represent calculations. However, using symbols alone, as pointed out previously (Steenrod et al., 1973; O'Halloran, 2005), is insufficient to communicate mathematical ideas and solutions in a school context. Other representations are needed, but because of the multi-semiotic nature of mathematics, this need presents another challenge. It is unclear in what ways different semiotic resources should be used and combined and what roles the different representations, such as natural language, i.e. words and sentences, play. After reviewing the above research, we can conclude that the answer to these questions cannot be found there.

Proof writing was the only type of mathematical writing explicitly concerned with rules or advice that students can use when ‘learning to write’ and with writing to communicate with someone (other than supplying the teacher with something to assess). Coincidently, as was mentioned in the introduction, Stylianides (2007) also introduces proof writing as a type of writing that offers objective rules and variation with context, such as age. As a consequence of the lack of agreement on how to define and measure quality in students’ writing within the discipline of mathematics, we argue that for students to learn how to write, teachers will have to create teaching that combines the two conceptualisations of writing in which (i) there are universal rules to consider but at the same time (ii) an opening to create a local discourse where the meaning is negotiated and norms for what is considered good writing is developed through a genuine interest in being understood. An opportunity to do this is to be inspired by proof writing. In our call for more research on the issue of students’ mathematical writing, we want to put forward the idea that the four aspects of quality described in the proof-writing studies described above could be taken as a point of departure for a study of teaching mathematical writing in a problem-solving context in school. Adjusted to such a context, the four aspects could be expressed and elaborated into aspects of mathematical writing more generally, as follows:

When reporting on problem solving, students should.

  • Define assumptions and assign variables, i.e. clarify the relevant information to solve questions given.

  • Produce a coherent narrative, including relevant calculations (semantic issues).

  • Use correct language, representations, and mathematical symbols (syntax issues).

  • Attend to what is appropriate in the context

In texts where students have attended to the above, the quality of their mathematical writing could be related to how clear, unambiguous, and efficient their communication is in relation to both the classroom context and to universal mathematical and linguistic rules. In this way, the two conceptualisations of mathematical writing are combined, and writing can become an object of learning separated from other learning objects in the mathematics classroom and given the attention it deserves.