developing mathematics written communication

Developing mathematics written communication through expository writing supported by assessment strategies1

Leonor Santos Sílvia Semana

Leonor Santos, Institute of Education, University of Lisbon, Portugal, [email protected] Sílvia Semana, PhD student, Institute of Education, University of Lisbon, Portugal, [email protected]

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Santos, L., & Semana, S. (2015). Developing mathematics written communication through expository writing supported by assessment strategies. Educational Studies in Mathematics, 88(1), 65-87. (DOI: 10.1007/s10649014-9557-z)

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Postal address of the corresponding author: Leonor Santos Instituto de Educação da Universidade de Lisboa Alameda da Universidade 1649-013 Lisboa Portugal

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DEVELOPING MATHEMATICS WRITTEN COMMUNICATION THROUGH EXPOSITORY WRITING SUPPORTED BY ASSESSMENT STRATEGIES

ABSTRACT This study concerns expository writing in mathematics as well as the contribution of assessment strategies to the development of mathematics communication. We studied four 8th grade students (aged 12 to 13) working in a group, in order to perform three expository writing tasks, which were assisted by feedback and the use of supporting assessment documents (a script and an assessment criteria grid in the form of rubrics). Our findings suggest that there was some positive development in the students’ expository writing throughout the study, particularly regarding interpretation and justification. The group of students were able to properly interpret what was asked of them, with reasonable levels of correction and completeness. Throughout the tasks, the group gradually included more relational justifications, instead of vague statements, rules or procedural descriptions. Students used multiple types of representation. In general, further explanations were made through verbal language. The assessment strategies contributed to such development, despite some prevailing limitations.

Keywords: expository writing; mathematical communication; assessment for learning; feedback; assessment criteria.

1. INTRODUCTION Communication in mathematics is recognized as an important aspect of mathematics learning and it includes sharing and explaining ideas, orally and in writing (NCTM, 2000). Although these two forms of communication are important, writing may be more effective than the use of thinking as it allows processes for supporting a metacognitive framework (Pugalee, 2004). The interest in written communication is not new in the international literature on mathematics education research. Previous studies reveal that written communication contributes to the learning of mathematical concepts and procedural knowledge (Porter & Masingila, 2000; Stonewater, 2002), to the development of problem solving skills (Borasi & Rose, 1989) and metacognitive abilities (Pugalee, 2001) and to the perception of mathematics and mathematical activity (Clarke, Waywood, & Stephens, 1993). Nevertheless, "communication in mathematics is not a simple and unambiguous activity" (Clarke et al., 1993, p. 249). Students tend to write in an unclear and vague way (Stonewater 2002) and require an extended experience in mathematics classrooms (Atieri, 2010; Davidson & Pearce, 1990). Moreover, Porter and Masingila (2000) questioned if the learning is actually determined by the writing or simple the time spent on thinking that the writing demands? Also Pugallee (2001; 2004) recognized that more studies concerning writing in mathematics are required and defended the need to study students' writing in detail to better understand the contribution of writing in promoting high-order thinking.

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Students' writing is also important for the teacher, because it may constitute an important resource of information about the way students think (Pugalee, 2001). When student thinking is written in students' words it allows teachers to adjust their instruction and to give the adequate support to each student (Back, Mannila, & Wallin, 2009). In other words, it allows teachers to develop a formative assessment, an assessment that is intended to enhance students’ learning (Black & Wiliam, 2006; Wiliam, 2007). This type of assessment considers assessment a learning opportunity and emphasizes students as active actors in the assessment process (NCTM, 2000). This perspective demands a proper integration between assessment, teaching and learning (Pinto & Santos, 2006) and the development of strategies that may regulate all the processes involved. The present study focuses on the development of mathematical written communication through expository writing supported by formative assessment strategies of the teacher's responsibility. Specifically, it considers four students of the 8th grade (aged 12 to 13) working in a group, with the aim: (i) to understand if and how the main features of the students' expository writing evolve throughout the study; (ii) to understand how the use of assessment strategies, namely feedback and sharing supporting documents with students, contribute to the development of students’ expository writing. 2. MATHEMATICS COMMUNICATION Learning is performed through a process of mediating activity (Vygostky, 1987). It is through language that the meanings for the different activities are negotiated, activities in which the students are involved, and in which they learn and recognize themselves as learners. The transformation of social language into an individual language is an essential requirement for learning. This passage requires the support of an effective communication process, a mediating process between subject and object. Communication is an indispensable dimension of this mediating process. It is a way to learn mathematics and a skill to be developed: “[students] communicate to learn mathematics, and they learn to communicate mathematically” (NCTM, 2000, p. 59). According to Sfard (2009, p. 175): learning mathematics means modifying one's present discourse so that it acquires the properties of the discourse practiced by mathematical community. Such change may be attained by a straightforward addition - by extending the vocabulary, by developing new routines or by producing and endorsing new narratives. Particularly, "communicating mathematical thinking and reasoning is an essential part of developing understanding" (NCTM, 2009, p. 3). Communicating in writing can be especially efficient in the development of students' mathematical understanding. It is a process that helps students to understand, to extract meaning, and to develop complex ideas (Chapin, O'Connor, & Anderson, 2003). Students are expected to present and explain their methods for solving problems and to justify their reasoning and the results of their thinking increasingly clearly, coherently and progressively in a more formal way, to themselves or to others,. It “can lead to the development of higher cognitive functions, including critical thinking, sound reasoning, and problem-solving” (Albert, 2000, p. 109). Several studies in this area support

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the idea of writing as a tool that can improve mathematical understanding (Albert, 2000; Borasi & Rose, 1989; Pugalee, 2004; Shield & Galbraith, 1998). In the present study, it is assumed that communication in mathematics includes interpretation, justification and the use of representations. Students create understanding based in their own previous experience and the individual interpretation of each situation (Cobb, Yackel, & Wood, 1992). The identification of the task goal, or in other words the correct interpretation of what is asked (Winne & Hadwin, 1998) is the first step for students to be able to communicate mathematical ideas with themselves and to others. Justification is an important component of students’ mathematics communication. The types of justifications presented by the students can give information about their understanding. However, students seem especially concerned with producing correct solutions (Sanchez & Sacristan, 2003) rather than with justifying their solutions. Previously, Back et al. (2009) studied students’ justifications in exams’ answers in high school mathematics and described the types of justifications found. The evidence showed that the type of justification changed throughout the course (broad/vague justifications decreased and relational ones increased), as a result of the nature of the task and of the perceived level of difficulty. Presenting justifications implies the use of representations (Duval, 2006; Golding, 2008). "Representations can help students organize their thinking (...) and make mathematical ideas more concrete and available for reflection" (NCTM, 2000, p. 68). Representations can be grouped into systems of internal representations (mental representation system) and external ones (semiotic representations). These representations, in mathematical communication, may have different types, such as: spoken or written language; active representations (using simulations and / or manipulative materials), iconic (using images more or less structured such as drawings, schematics, diagrams) and symbolic (symbolic mathematical language) (Bruner, 1999). As students evolve in their mathematical learning, it is expected that they increasingly use conventional forms of representation instead of non-conventional ones (NCTM, 2000). For a deep understanding, it is necessary, not only to know a representation and to use it appropriately, but also to be able to know a variety of representations and to be able to move flexibly between them (Duval, 2006). Different kinds of writing may take place in a mathematical classroom, including expository writing, which often pertains to how to perform a mathematical procedure or to explain why a given mathematical outcome occurred. Shield and Galbraith (1998) studied this type of writing in students of 8 th grade math classes. They concluded that the statement of the goal of the task was normally presented by the students, whereas justifications were presented in a smaller number, revealing a limited style of expository writing and a restricted perspective of what constitutes doing mathematics. 3. ASSESSMENT FOR LEARNING Assessment for learning, or formative assessment, is characterized by day-by-day classroom practices involving elicitation and interpretation of evidence about students' learning and the ongoing use of this information to make founded decisions about teaching and learning, in order to support students’ learning.

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These decisions are commonly made by the teacher, but the students themselves may also be involved in such decisions, for example, through self-assessment (Black & Wiliam, 2009; Brookhart, Andolina, Zuza, & Furman, 2004). Self-assessment, as an internal process of regulation of students’ own thinking and learning, is vital for an effective learning (Nunziati, 1990). Feedback is a key element of assessment for learning and a first step to promote self-assessment. It is a deliberated dialogue that aims to help students to overcome their difficulties in the approach between the intended and the performed (Sadler, 1989). However, not all feedback is effective. “To be powerful in its effect, there must be a learning context to which feedback is addressed” (Hattie & Timperley, 2007, p. 82). Feedback of a descriptive nature (Gipps, 1999), focusing on the task, process or self-regulation (Hattie & Timperley, 2007), whilst providing information about what needs to be improved and how students may do it is potentially more conducive to learning (Santos, 2008). Interrogative or mixed feedback tends to be more effective than merely affirmative feedback (Santos & Pinto, 2009). Feedback should be given during the development of the task, before its rating (Santos & Dias, 2006). Another important aspect of effective assessment is the use of assessment criteria. However, in order to be considered as a reference to students’ self-assessment, it needs to be legitimate from the students' standpoint (Nunziati, 1990). In other words, the criteria must be appropriated by the students. However, this is quite difficult since the meaning students give to the assessment criteria may be different from that given by the teacher (Vial, 2012). So, it is necessary to create opportunities for students to understand the criteria in the context of their own work. Besides feedback and sharing the assessment criteria, other assessment strategies may be used to promote learning, such as the development of tasks in two stages (Santos, 2002). 4. METHODOLOGY 4.1. Method The chosen methodology was qualitative and interpretative in nature (Goetz & LeCompte, 1984). We studied the expository writing of a group of four students of an 8 th grade class, working collaboratively, in the development of three expositions (we used the term expositions for students’ written productions as result of expository writing tasks). This study involved a class of 24 students of the 8 th grade (aged 12 to 13) from a school in the North of Portugal and a mathematics teacher, who welcomed the researcher (the second author of this paper) into the classroom. From this class, four students – Maria, Rute, Duarte and Telmo – were selected by the researcher to form the group that was the subject of study. The composition of the group remained unchanged throughout the study. To select the students, three criteria were taken into consideration: (i) ability to interact with others; (ii) diversity of gender; and (iii) diversity of mathematical performance. With these criteria we pretended to ensure that the participants were good informants and to guarantee the diversity of the class, particularly with regard to written communication (Table 1). Information about participants' mathematical performances and initial written communication skills were given by the teacher.

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Maria

Table 1: The participants´ performance in mathematics Learning disabilities and poor performance in written communication.

Rute

Good performance in mathematics, including in written communication.

Duarte

Good performance in mathematics and reasonable performance in written communication.

Telmo

Revealed skills which are not very consistent, in particular in written communication.

Data was collected during the academic year 2007/2008, mainly through: (i) participant observation (Lessard-Hébert, Goyette, & Boutin, 2005) of ten lessons (audio recorded) in which the students developed three expository writing tasks and/or assessment strategies were implemented; (ii) collection of two versions of each written exposition, done by the group of students; (iii) four semi-structured interviews (Fontana & Frey, 1994), audio-recorded, with each participant student, one at the beginning of the school year and the others after completing the second version of each expository writing. These interviews were intended to understand students’ perspectives regarding the process of development of expository writings and the role of assessment strategies in that process. The following code will be used to identify excerpts from each of the interviews: I "number of the interview", "student name" (e.g., “I2, Maria” corresponds to the second interview with Maria). The transcriptions and students’ written expositions were analyzed in Portuguese and then translated to English. The analysis of the written expositions of the working group (EW1, EW2, EW3) took into account: 1. The interpretation of the goal of the task, considering: - the goal statement (Shield & Galbraith, 1998): Inclusion (identification of the general goal of the mathematical task that is subject of the writing); Completeness of the information included; - the language use: Format (Transcription of the goal statement in the assignment statement or Rewritten using their own words or); Precision of Language. 2. The justifications presented, considering: - the type of justification (adapted from Back et al., 2009): Vague/broad statement (unclear or uninformative explanation); Rule (the exclusive use of a rule, algorithm or definition); Procedural description (explanation of what is done in a certain step, without explanation of why the step is valid); and Relational justification (explanation for why a step is valid, including or not the explanation of what is done in a certain step, suggesting a relational understanding) (Note that each justification can only be of one type); - the correctness and the completeness of the justification. 3. The representations used, considering:

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- the type of representation: Verbal language (natural language, students words and mathematical terminology), Iconic representation (a scheme or drawing), and Symbolic representation (numerical and/or algebraic symbols); - the precision and the completeness of the representation. Regarding the interview data, the audio recordings were transcribed and the relevant parts of the transcripts were selected in order to capture the perspectives of the students regarding: (i) the relevance they attributed to each of the assessment strategies adopted, (ii) the use they gave to the assessment strategies in writing the expositions and the reasons for using them (or not). Written feedback was analyzed using the records written in the first version of the expositions and was categorized taking into account its contribution to the process of writing and its effects on students' written expositions (in particular, whether if it was effective or not). A similar analysis was applied to oral feedback, transcribed from audio recordings of classes. 4.2. Mathematical tasks Taking into account that demanding writing tasks may constitute a favorable context for the development of mathematical communication, moving progressively from informal to more formal explanations (Levenson, 2010), three high-level cognitive mathematical tasks have been proposed to students as prompts for expository writing. The first task intended to: (i) promote students’ reflection about the meaning and implications of the Pythagorean Theorem; (ii) evoke some geometric concepts and procedures; and (iii) promote the development of mathematical reasoning, by appealing to the formulation and testing of conjectures and argumentation. Students were not expected to formulate a general conclusion.

The second task involved the use of the Pythagorean Theorem in space and was proposed after a lesson in which it was addressed by determining the diagonal of a parallelepiped. The task intended to: (i) call for the use of Pythagorean Theorem to calculate distances in space; (ii) to evoke some concepts (e.g. circular sector, radius, height of a cone) and mathematical procedures; and (iii) to promote problem solving skills and generalization of the solving process.

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The third task was a numerical game – Tudo ao Monte – in which two players, alternately, take one of three balls out of a set of several balls. The player who takes the last ball loses the game. Students were asked to answer several questions in order to identify a winning strategy and which player has an anticipated advantage, when the total number of balls at stake is 9, 21 or 22. This task was proposed following a set of lessons focused on the mathematical topics of sequences, greatest common divisor and least common multiple of two numbers. Students had to identify the winning strategy, starting by exploiting a situation similar to the original game, but more elementary (9 balls); then explore the case of 21 balls of paper (original game); and finally explore a situation that implied changing the winning strategy (22 balls). The task was intended mainly to foster problem solving skills, mathematical reasoning and argumentation. The writing tasks, which included performing the mathematical tasks and writing the expositions, were developed within a period of 90 minutes, in the classroom and in small groups of three or four students, including the group of students that was studied (constituted by Maria, Rute, Duarte and Telmo). Between the first and the third exposition four months passed. The three expositions were developed in two stages: the groups of students wrote a first version to which was given written feedback and then they developed a new version. 4.3. Assessment strategies In this study, formative assessment strategies included sharing supporting documents with students (script and assessment criteria grid) and giving oral and written feedback. The students had never prepared any written exposition that included an interpretation of the goal of the task, a justification, and the use of multiple representations, but they were already familiar with the use of scripts for guiding other types of productions. They also had never been exposed to specific assessment criteria for performing mathematical tasks. In terms of written feedback, students were used to receive this type of feedback on assessment tests carried out in two stages (as with the expositions in this study). The script and the assessment criteria grid were created to support the writing of the expositions. Their implementation in the classroom was planned by the researcher and the math teacher, in collaboration. The script explained what was expected of the students in expository writing and why it was important (see the Appendix I for an example of what was included in the script). It also gave indications about what aspects the students’ written expositions should be focused on. The assessment criteria grid (in the form of rubrics) presented descriptors for four levels of performance (not ratings) for each criterion, regarding: structure of the exposition and presentation of the written text; solving process; prior knowledge; description and

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justifications; and mathematical language (see the Appendix II for an example of the assessment criteria grid concerning description and justification). This grid was shared with students and used as basis for a prior assessment experience involving students, previous to the writing of the first exposition. Written feedback was added to the first versions of students' expositions. It was prepared by the researcher, taken into account the script and the assessment criteria. It was intended to promote students' reflection. Through clues to future action, it focused on aspects to improve. In addition, feedback did not include the correct response or the correction of errors, took the interrogative form and tried to use an accessible language. In some cases, written feedback also identified positive aspects of the work presented, so that those aspects were consciously recognized by students and their confidence was promoted. Oral feedback was given, either by the researcher or the mathematics teacher of the class, mainly in the second stage of writing, when students expressed doubts or difficulties concerning written feedback. 5. RESULTS The students' written expositions were analysed considering the interpretation of the goal of the task, the justifications presented and the representations used by the students. 5.1. Interpretation During the development of the first version of EW1, students consulted the script to determine how they would structure their exposition. In particular, by consulting the script, students realized that they should present the task and explain its goal: Rute: Maria:

Oh Duarte, what do you think I should write in the introduction? (...) It [the script] says to present the task and to indicate what its goal is. The aim ... the purpose of it... (Lesson 3)

After this dialogue, students wrote a goal statement in their exposition, rewriting an assumption that was given in the mathematical task: “Development of the Pythagorean Theorem and its introduction in other geometric figures” (v1, EW1). We coded this first version as incomplete and lacked precision in the language. Written feedback was added to this version of the work, in order to clarify the goal statement "What does this mean? Try to write a complete sentence, rather than messages like telegrams" – and to explain the Pythagorean Theorem – "You also refer the Pythagorean Theorem, but it is unclear if you know what it means. Why don’t you explain?". This feedback was used by the group of students, in order to develop the final version of their exposition. They explained more accurately and clearly the goal of the task and established a link to prior knowledge, this time not only referring the Pythagorean Theorem but also showing that they understood the corresponding relation: In this work, it was proposed to us to develop the Pythagorean Theorem. It had been presented to us a problem: Constructing geometric figures on the sides of a right triangle of the areas of the geometric figures on the legs are equal to the area of the figure on the hypotenuse? (v2, EW1) One of the students stated that the written feedback helped them to explain/clarify the goal of the task, revealing their understanding: “this comment... also helped us a lot, because the teacher said that the

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messages were like telegrams and we should explain the purpose of the task and we did it in the second stage” (I2, Rute). During EW1, students did not consult at any moment the assessment criteria grid. During EW2, the situation was reversed: students did not use the script, but instead they used the assessment criteria grid (only during the first version of the written exposition). As the students explained, they were already familiar with the script. Therefore they already knew which aspects they should focus on. In the first version of the EW2, students presented the main elements proposed by the script. Students considered more relevant, at that moment, the use of the assessment criteria grid to guide the writing of the first version of their exposition, and particularly the introduction: "The script ... we have not used because we already knew how [the exposition] should be organized, it should have the introduction..." (I3, Maria); "we did not use the script because we had (…) the criteria and they were more useful...” (I3, Telmo). In accordance with the supporting documents, in the first version, students included, in the introduction, a goal statement, rewriting an assumption that was given in the original mathematical task: In this exposition we will talk about the work with the cones. This task asks us to investigate the height of the cones that were constructed in a circle. For this we had to draw a circle and divide it into three equal parts. Then we wanted to find out the cone’s height in cm. (v1, EW2) Despite some inaccuracies of language (for example, saying that the cone is constructed in a circle) feedback was not added to this part because it was considered already acceptable. Feedback focus on other aspects less successful in other parts of the written exposition. Finally, regarding EW3, during the development of the first version, students consulted the script to become aware of the aspects they should include in their exposition, namely in the introduction: "Look, what we put down in the introduction? Here [the script] says that we must indicate the goal of the task" (Lesson 9, Rute). Due to the fact that some time had passed since the last expository writing (EW2), students felt the need to remember what aspects should be included in their exposition: “We used the script because we had already forgotten some things, because of the holidays and everything. Thus, we remembered what we should put down” (I4, Maria). In accordance with the script, students included, in the first version of the introduction, a goal statement, once again, rewriting an assumption that was given in the original task: "This work has the goal to find the best strategy to win the game Tudo ao Monte, answering several questions and justifying" (v1, EW3). So, the students explained that the goal was to find a strategy to win the game, but they did not explain the game itself. Written feedback suggested students add such explanation: “Maybe you could explain how the game Tudo ao Monte works”. As the students uncovered challenges in completing this part of the task, oral feedback was given, recommending the reading of the mathematical task statement/formulation: "See what it says in the task statement/formulation. Maybe it's easier, to explain [how the game works], is it not?". This suggestion helped students to explain how the game works: “The game Tudo ao Monte is about a

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certain number of balls, in which two players take, alternately, one of three balls. The player who takes the last ball loses” (v2, EW3). The assessment criteria grid was consulted by the students only after completing the first version of EW3, apparently to confirm that their production was adequate, in particular regarding the introduction. The group decided that a more careful analysis of the assessment criteria could be advantageous to improve their exposition, but it would require more time than expected: “First, the work should be well structured, then the results well explained and only then we should improve with the criteria, but we did not have much time ... and we could not do everything” (I4, Duarte). In summary, in all expositions, the group described the goal of each task in their own words, reveling understanding as result of an interpretation process. Considering only the first version of the expositions, it is even possible to observe a positive evolution in the presentation of the goal statements (Table 2). In particular, the inaccuracies have been reduced and the incompleteness in the third exposition did not reveal a faulty interpretation (the group of students described the goal statement, but they have difficulties to explain the game by their own words). When receiving feedback, from the first to the second version, the group improved the statement of the goal, supplementing it, and the language used, in terms of precision. Supporting documents have been used by students interchangeably, according to their needs and time available, as guides for their writing, namely for the first versions of the expositions. Table 2: Goal statement and language in expository writing Goal statement Language Included Rewritten Incomplete Lack of precision 2nd version Included Rewritten Complete Accurate 1st version Included Rewritten Complete Lack of precision 2nd version -

Expository writing 1st version EW1 EW2

EW3

1st version 2nd version

Included Incomplete Included Complete

Rewritten Accurate Rewritten Accurate

5.2. Justification The first version of EW1 reveals several limitations with regard to the type of justifications and the level of completeness. For example, concerning the exploration with equilateral triangles, the group of students only presented a procedural and incomplete description of the exploration process and a particularly vague statement for explaining the use of the compass: We have completed the first task; we started by drawing a right-angled triangle, then with the help of the compass we drew around it (at each vertex) creating three equilateral triangles, doing so because the ruler didn’t allow us to obtain an equilateral triangle nor a good layout. Through this we determined the area of the triangles. (v1, EW1)

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Additionally, students presented the corresponding geometric construction and some related, but incomplete, calculations (Fig. 1).

Fig. 1: Equilateral triangles construction and calculations (v1, EW1)

Although it is not possible to establish a direct relation between the type of justifications presented and the use/non-use of the assessment criteria, the absence of relational justifications in the first version of EW1 may be associated with the fact that students have not consulted the assessment criteria grid. Students recognized that the use of the criteria might have been useful for them to improve their explanations and include relational justifications: “They [the criteria] might have helped. Maybe I had better explained the main ideas of the activity...” (I2, Rute); “Maybe I had (…) presented [relational] justifications... I do not know, but I do not think it is important” (I2, Duarte). Special attention should be given to the last statement of Duarte, in which the student says that presenting relational justifications was not important for him. In fact, possibly as a consequence of the non-use of the assessment criteria, after completing EW1, students still seemed to value the presentation of the solving process and solutions in expositions, instead of the presentation of relational justifications: [It’s more important] the steps we have done [to solve the task] and the conclusions we have taken from the exercises. I think that is the most important. Explaining how and why can always be improved and be done in other ways, but what we really have done is the more important. (I2, Duarte) Even so, in the final version, there was an improvement in the students’ justifications apparently related to the feedback given. Written feedback asked students to explain what they had done and why they had done it: “How did you get these values? What did you conclude with the calculation of these areas? What relation did you establish?"; "Try to give better explanations for the geometric constructions and calculations presented and clarify yours conclusions". In the final version, students explained how they had

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determined the area of the triangles and why that procedure was valid (they established a link to prior knowledge, referring to the formula to determine the area of a triangle), and they explicitly made the relationship among the areas of the equilateral triangles, and that each had a side which formed the rightangled triangle: We determined the triangles' areas. We know how to find the area of a triangle: (base x height) / 2. We measure the height and the base, multiply them and then divide them by 2 (and we did it to the three triangles). We concluded that the sum of area A and area B is equal to area C. (v2, EW1) Students also presented, in a correct and complete form, the respective calculations (Fig. 2), and concluded a relational justification concerning the exploration with equilateral triangles. In addition, students identified some less accomplished aspects of the first version of their exposition (Fig. 2).

Fig. 2: Equilateral triangles (v2, EW1)

In the first version of EW2, students mainly used relational justifications. For example, in the first exploration (regarding the circle of radius 6 cm), students included a description of the procedure used to divide the circle in three equal circular sectors and explained why that procedure was valid, establishing a link to a prior knowledge, referring to the measure of a full angle: We drew a circle with 6 cm radius. To divide the circle into three equal parts we know that the angle measure is 360º (so 360º / 3 = 120º). With the help of the protractor we measured 120º on a radius three times and we joined the dots and got three equal sectors. (v1, EW2) Just in a step of EW2, the justification presented was not a relational one, but rather a rule, since students have presented the calculations to determine the perimeter of the original circle, the perimeter of the cone’s base, and the radius of the cone’s base, but they did not provide meaning to those calculations, in order to reveal a relational understanding. In particular, they did not explain the calculations, did not distinguish

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between the two circles referenced (the original one and the cone’s base) and did not present units of measurement (Fig. 3):

Fig. 3: Calculations to determine of the radius of the cone’s base (v1, EW2)

Improvements were observed in the second version. Written feedback was given to the first version, supplemented in some cases with oral feedback. For example, regarding the rule presented in the first version, written feedback was given alerting students with regard to the less accomplished aspects and giving indications for future action: Why have you done these calculations? You mention the perimeter of the circle several times. Maybe it would be better if you explain which circle you are talking about in each case. At the students' request oral feedback was given in order to clarify the written feedback: Teacher:

Rute: Teacher:

Rute: Teacher:

You should try to explain the calculations presented and why you have done them. You presented these calculations, did you not? Why? What for? When? How? The teacher wants to know everything! I want to know everything, I do not ... Imagine that I am giving a lecture and I write something on the board and you ask me "teacher, what is that?" and I tell you "You want to know everything!" Oh teacher, but here we already know that this is the perimeter... You know, but you have to write what you mean, I am not going to take Rute home to tell me, right? (Lesson 7)

Students seemed to take into consideration the feedback, both oral and written, and in the final version of the exposition they described how they had determined the radius of the base of the cone. They clarified the purpose for the calculations presented; they referred to the formula to determine the perimeter of a circle; they explained which circle they were referring to; and they presented relational justifications, including explanations for what they have done: First we found the perimeter of the circle in question. Next, we divided the perimeter of this circle into three equal parts and we got the perimeter of the base of a cone. Knowing that to find the perimeter of a circle is 2π.r, we therefore know the radius is P/ 2π = r. And we obtained 1.9 cm. (v2, EW2) Students also took into account the written feedback addressed to the second exploration: "Do not forget (...) the last question in the task". Although students did not determine the height of the cone in terms of the

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radius x of the original circle, they presented an initial description of the strategy to be followed by establishing a parallel to the particular case already treated (Figure 4). However, a relational understanding of the situation was not totally evident, since the last passage (marked on figure 4) pointed out to difficulties in manipulating the algebraic elements involved and/or providing meaning to those elements. Furthermore, students did not complete the justification for this exploration, which seems related to the lack of time (students did not even conclude the process of exploration, as observed by the researcher). The presentation of relational justifications in the first version of EW2 may be associated, in any way, with the fact that students have consulted the assessment criteria grid for writing such product. After completing EW2, students changed, at least slightly, their opinion concerning what was important to include in expository writing, recognizing that explaining how and why was more important than they thought: “[It’s important] how we solved the problem and how we have justified the steps. It's not just putting the results, we have to explain” (I3, Rute); “[It’s important] the solving process and how we have explained it. It will be 45% for the response, 35% for the justification…” (I3, Duarte).

Fig. 4: Exposition for the second exploration (v2, EW2)

In the first version of EW3, students revealed a relational understanding by identifying, in a sustained manner, a winning strategy for each of the proposed situations, whatever the course of the game. However, the justification for why this strategy guarantees a win was carried out by exhaustion, taking into account all the possible moves of the opponent (Fig. 5). The process of justification was effective, but not so efficient. This may be explained by the fact that the students did not use prior knowledge to give mathematical meaning to the situation and to find alternative justifications. Establishing a link with prior knowledge in this case was, in fact, more difficult than in the past because there was not a straight relationship between the task and the mathematical contents previously studied. Regardless, the main

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limitations identified in the first version of EW3 were related to the level of correctness and completeness of justifications. For example, for the exploration with 21 balls at stake students included an imprecise statement for the winning strategy students, without further explanation beyond the diagram in Fig. 5: “the second player wins if he uses the strategy of taking a sum of four balls in the moves” (v1, E3).

Fig. 5: Diagram to support the wining strategy for the game with 21 balls (v1, EW3)

The final version of EW3 improved compared to the first version, concerning the level of correctness and completeness of justifications. Written feedback was given in order to lead students to clarify what strategy the second player should follow to win the game: “What do you mean by taking a sum of four balls in the moves? What moves are you referring to?"; “You should provide a description to accompany and explain this diagram, in order to clarify which strategy allows the second player to win the game”. Given the difficulties shown by the students, oral feedback was given to clarify what was intended and to guide the students’ actions in order to respond to the written feedback: Teacher:

Rute: Teacher:

Here you say the second player is in the lead if he/she takes a sum of four balls in the moves, right? What do you mean? What does a sum of four balls mean? The number of balls that were taken must be four. And what does the second player need to make it happen? (Lesson 10)

In the final version, students presented, correctly and completely, the strategy that the second player should use to win the game with 21 balls: If the first player takes 1 ball, the 2nd player takes 3 balls; if the first player takes 2 balls, the 2nd player takes 2 balls; if the first player takes 3 balls, the 2nd player takes 1 ball, always making a total of 4 balls. (v2, EW3) This explanation, combined with a new diagram presented in the second version of EW3 (corresponding to the diagram in Fig 5. but a more elaborated and clearer one, similar to the one in Fig. 8), represented an evolution in the quality (particularly, in terms of correction) of the relational justification presented.

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Although the assessment criteria grid have been consulted by the students only at the end of writing each part of the first version of EW3, the type of justifications presented in that version of exposition was already in accordance with the assessment criteria. In fact, by the end of the study, after completing EW3, the students, including Duarte, have recognized the value of “explaining how and why” in expository writing: [It’s important] the way we have explained the strategies and why the answers are true or not. Because it is no use to say that the first player has an advantage, we must explain how we arrived at this conclusion. (I4, Rute) [It’s important] how we have explained the problem (...) In the first stage we have presented the strategy, but it is not so ... well explained or justified. If I was not aware of the game, I would not realize what the strategy was. But in the second stage is better. (I4, Duarte) In summary, it is possible to identify a trend of evolution in the quality of justifications presented by students in their expositions by comparing the two versions of each exposition or the three expositions (Table 3). Comparing the two versions of each exposition, there was an improvement of the type of justifications, with the inclusion of more relational ones (namely in EW1 and also in EW2), and also on the level of correction and completeness of justifications. These changes were potentiated by the written feedback and supplemented by the oral one. Comparing the three expositions, particularly the first versions, there was also an improvement, with a progressive inclusion of relational justifications. This positive evolution may be confirmed partially with respect to correctness, but not in relation to the completeness (the first version was always incomplete). The use of the assessment criteria grid seems to have lead students to progressively value “explaining how and why” and to include more relational justifications in their expositions. Table 3: Justification type, correctness and completeness in expository writing Expository writing Type Correctness Completeness 1st version Vague statement Incorrect Incomplete Procedural description EW1 2nd version Relational justification Partial correct Partial complete EW2 1st version Rule Correct Incomplete Relational justification 2nd version Relational justification Correct More complete, but still incomplete (lack of time) EW3 1st version Relational justification Partial correct Incomplete 2nd version Relational justification Correct Complete 5.3. Representations For presenting and explaining the mathematical explorations resulting from the original mathematical tasks, students used multiple types of representation, combining verbal language with iconic and symbolic representations. In the first version of EW1, concerning the exploration with equilateral triangles, students used verbal language essentially to present procedural descriptions (see exemplar in section 5.2.) and used iconic and symbolic representations (see Fig. 1) to present the process of exploration, without justification. In the second version, students did not change the type of representations used (as it is shown before).

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Still in the first version of EW1, but now regarding the explorations with rectangles and semicircles, students used verbal language only to present their conclusions. Iconic representations – geometric constructions – and symbolic ones – numerical symbols – were also used, but they lacked meaning and did not represent any type of justification (see Fig. 6 for rectangles).

Fig. 6: Rectangles registers (v1, EW1)

In the second version, students improved and completed their iconic and symbolic representations and included verbal language (though with imprecision) to provide meaning to the previous representations, giving justifications (incomplete relational justification, including explanations for what was done). This is evident in the following transcription, as well in Fig. 7: We held the second exploration proposed, starting with making several rectangles around the right triangle. With the rectangles, we obtained the following sum: area y + area x ≠ area h

8 cm2 + 8 cm2 ≠ 34,2 cm2

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Fig. 7: Rectangles registers (v2, EW1)

The identified changes, from the first to the second version, concerning the representations used, and particularly the inclusion of verbal language to provide justifications, seems to be related to the focus of feedback given to the first version of this part of exposition (presented in section 5.2. as well as in Fig. 6). Although feedback did not directly refer to representations, it presupposed changes (in particular more verbal language and numerical/algebraic symbols) in such representations. Concerning the EW2, in the first version, students mainly used symbolic representations combined with verbal language (with some imprecision) to present relational justifications. For example, see the corresponding examples in section 5.2. The only remarkable change, from the first to the second version of EW2, was the use of additional verbal language to respond to the given feedback, providing more explanations to improve justifications – students transformed a rule into a relational justification and completed relational justifications (see examples of corresponding oral and written feedback and of students’ final exposition in section 5.2.). Once again, although feedback did not directly refer to representations, it required changes in such representations. In both versions of EW3, students used verbal language, essentially to identify the player that had the advantage and present a winning strategy, and used a combination of iconic and symbolic representations – a kind of diagrams, with numerical symbols (see for example Fig. 5) – to clarify the winning strategy and justify why it was valid. From the first to the second version, as in the previous expositions, students added some additional explanations in verbal language to respond to given feedback. But the most significant

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change was related to the diagrams used to justify the winning strategy. Concerning the exploration with 22 balls, in the first version, students used a diagram to illustrate the strategy that the first player used to win the game, but the diagram was clearly incomplete. Written feedback was given, on one hand, to praise the students’ work and, on the other hand, to lead them to finish it: "Good job, the diagram is well done! Why do not you finish it?". This feedback was helpful in the final version as the students finished the diagram (Fig. 8).

Fig. 8: Diagram to support the winning strategy for the game with 22 balls (v2, EW3)

In addition, the given feedback helped students to become aware of a positive aspect of their work, since they made a similar diagram for the exploration with 21 balls to replace the other diagram (Fig. 5) that they had presented in the first version. In the three expositions, students used multiple types of representation, combining verbal language with iconic and symbolic representations (Table 4). Particularly, the use of iconic representation, at least in the first version, may be associated to the fact that the script, besides suggesting the use of verbal language, explicitly refers the possibility of using tables, graphs or diagrams. The use of symbolic representation, in turn, seems more related to the mathematical tasks used as prompts for the expository writing, which evokes the use of numerical and/or algebraic symbols: while in EW1 and EW2 students used algebraic symbols, in EW3 students only used numerical ones. In general, from the first to the second version of expository writing did not imply changing the type of representations, but rather improving such representations (in terms of completeness and precision) and adding more verbal language to provide further explanations and give meaning to the remaining representations. These variations in representations, from the first to the second version of expository writing, were potentiated by feedback, although, in general, it did not directly refer to representations.

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Table 4: Type of representations used in expository writing (goal statement and justifications) Expository writing Goal statement Justifications

EW1 EW2

EW3

1st version 2nd version 1st version

Verbal Verbal Verbal

2nd version

Verbal

1st version

Verbal

2nd version

Verbal

Verbal, Iconic and Symbolic Verbal, Iconic and Symbolic Essentially Verbal and Symbolic (with Iconic) Essentially Verbal and Symbolic (with Iconic) Verbal, Iconic and Symbolic (only numerical symbols) Verbal, Iconic and Symbolic (only numerical symbols)

6. DISCUSSION The present study was intended to study the mathematical written communication of a group of students of the 8th grade, through expository writing supported by formative assessment strategies. Mathematical written communication was analysed regarding interpretation, justification and representation, through written expositions, based in three different mathematical tasks. Considering that no one can describe and explain (Shield & Galbraith, 1998) without understanding what was asked, it is important that students understand the mathematical task proposed to them. The results evidenced that all the first versions of expository writing included a goal statement in the introduction. Students have rewritten an assumption given in the mathematical task. Although not always presenting complete information and using accurate language in the first versions, the group of students was able to properly interpret what was asked. This is coherent with the findings of Shield and Galbraith (1998) and deserves to be stressed that the progressive importance that has been given, in all teaching situations, to the step devoted to understanding the task (Tardif, 2007). Justifications used in students’ expository writing were diversified albeit showed a tendency towards including more relational justifications (in place of vague statements, rules or procedural descriptions), with reasonable levels of correction and completeness. Students tend to choose the level of elaboration and detail of their justifications based on the difficulty level of the task at hand (Back et al., 2009). The group of students used multiple types of representation. In all expository writing it can be found verbal language, iconic and symbolic representations. The most noticeable change that occurred from the first to the second version was the increase of the use of verbal language to add further explanations. This may be interpreted as the students were more comfortable with this kind of representation (Stonewater, 2002). The presentation of relational justifications always required the use of verbal language, along with the use of iconic and symbolic representations. An additional use of verbal language in the second version of the expositions was, generally, associated to an improvement in the quality of justifications presented (more complete, accurate and with relational justifications). Although it is not possible to identify an evolution in the quality of representations (in terms of completeness, precision…) used by the group of students, they

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generally used representations that allowed them to reveal understanding of the goal of the tasks (using verbal language) and to explain how and why to proceed to perform the tasks (presenting relational justifications). The assessment strategies used to support the students’ writing in a context of group work seemed to be useful to the development of expository writing. In particular, written feedback given to the first versions helped the group of students to improve the final versions of all expositions, regarding the aspects identified as least accomplished in the first versions. However, there were some examples of students’ expositions to which feedback was not provided, in particular concerning the goal statement of EW2. This decision was due to the fact that the teacher had to be selective, focusing only in the most relevant aspects of the learning process in a certain moment. The criterion is not the exhaustiveness, but it is the aspects to focus on (Veslin & Veslin, 1992). The effectiveness of the feedback provided in this study seems to be associated with the fact that it had two main characteristics: to include questions to orient the students' work and clues to the next action (Santos & Pinto, 2009; Wiliam, 2007). The oral feedback tends to be more effective than the written one due to the fact that it happens in the moment, providing an interactive regulation, and it can be applied to any case and developed to the intended level (Santos & Pinto, 2010). Increasingly, the group of students were able to interpret the supporting documents. They were appropriating the functions of each document and giving them a proper use, according to their needs. The script helped them to structure their presentations and to become aware of the aspects they should focus on. The assessment criteria grid contributed to the students' knowledge of what was expected in a high quality writing (Lim & Pugalee, 2004). Note, however, that this awareness did not come immediately. Students felt no need to use the assessment criteria grid in the first expository writing and did not appropriate all the assessment criteria in the first moment, which is in accordance with the results of other studies. It means that implementing learning experiences is required to develop mathematical writing (Clarke et al., 1993). It is necessary to note that due to the high cognitively demanding mathematical tasks proposed to the students as prompts for expository writing, in addition to the kind of mathematical tasks not being familiar to the students, we decided to work in groups. Consequently, students’ expository writing was analyzed as a product of a group of students. According to it, all the assessment strategies were directed to the group of students and not to any student in particular. It is in this context that the results of the present study have to be considered. We are conscious that the quality and the evolution of the expository writing may not have received the same level of input from each student and the interactions between students may have also given an important contribution to the performance of the group of students. More research is needed to understand how the individual written mathematics communication of each student can be developed through expository writing, accomplished with assessment strategies used to foster students’ expository writing. 7. CONCLUSION REMARKS

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Our findings evidence that, in a context of group work, the combined action of the expository writing experience and the implementation of the formative assessment strategies (written and oral feedback and supporting documents) contributed to the elaborateness of students’ expository writing, particularly regarding interpretation and justification, when we compare the first and second versions of each expository writing. Similar results may be confirmed, in general, when we compare the first version of the three expository writings. These findings contradict the ones of Shield and Galbraith (1998). The authors recognized little development in the elaborateness of the students’ writing throughout the study and found similarities between the students’ writing and the writing of students' textbooks (algorithmic aspects of mathematics). One possible explanation for the divergent results is the difference between the aims of the two studies. While Shield and Galbraith (1998) study did not aim to develop students’ expository writing in mathematics, the present one had this main pedagogical aim. The importance of the intentionality of the teacher gives an additional relevance to this study. Although the results cannot be generalized, given the methodological options, this study provides clues about how teachers may support students’ writing, using formative assessment strategies, in order to stimulate the development of mathematical communication and a deeper understanding of mathematical ideas. These contributions may be important if we take into account that, in general, teachers do not integrate writing in their common practice (Atieri, 2010), do not always recognize the importance of creating learning opportunities for students to develop writing communication (Ntenza, 2006), and assessment strategies to help students to learn mathematics are not very usual (Wiliam, 2007). 8. REFERENCES Albert, L. R. (2000). Outside-in – Inside-out: Seventh-grade students’ mathematical thought processes. Educational Studies in Mathematics, 41(2), 109-141. Atieri, J. (2010). Literacy + Math = Creative connections in the elementary classroom. Newark, DE: International Reading Association. Back, R., Mannila, L., & Wallin, S. (2009). Student justifications in high school mathematics. Proceedings of the Sixth Conference of European Research in Mathematics Education (pp. 291-300). Lyon: France. (http://ife.ens-lyon.fr/publications/edition-electronique/cerme6/wg2-12-back.pdf) Black, P., & Wiliam, D. (2006). Assessment for learning in the classroom. In J. Gardner (Ed.), Assessment and learning (pp. 9-25). London: SAGE Publication. Black, P., & Wiliam, D. (2009). Developing the theory of formative assessment. Educational Assessment, Evaluation and Accountability, 21(1), 5-31. Borasi, R., & Rose, B. J. (1989). Journal writing and mathematics instruction. Educational Studies in Mathematics, 20,347-165. Brookhart, S., Andolina, M., Zuza, M., & Furman, R. (2004). Minute math: an action research study of student self-assessment. Educational Studies in Mathematics, 57, 213-227. Bruner, J. (1999). Para uma teoria da educação. Lisboa: Relógio D'Água. Chapin, S., O'Connor, C., & Anderson, N. (2003). Classroom discussions. Using math talk to helps students learn. Sausalito: Math Solutions Publications. Clarke, D., Waywood, A., & Stephens, M. (1993). Probing the structure of mathematical writing. Educational Studies in Mathematics, 25, 235-250.

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Appendix I

Appendix II Assessment Criteria Grid for Expository Writing (Example for Description and justification)

The group [of students]…

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