Introduction

Mathematics educators strive to teach productively, to support students’ learning, and powerfully, to develop students’ positive identities as mathematics learners (Herbel-Eisenmann et al., 2013, 2015). Rough draft math [RDM] (Jansen, 2020a, 2020b; Jansen et al., 2016; Rathouz et al., 2023; Thanheiser & Jansen, 2016) is one approach to mathematics teaching that has the potential to achieve these goals. RDM occurs when teachers invite students to publicly share their in-progress thinking, position students’ initial thinking as drafts worth understanding, invite students to workshop their draft thinking, explicitly encourage students to revise their thinking, and encourage students to reflect on how their thinking grew and changed. Identifying how teachers enact RDM, including variations in their enactments, can help teacher educators support more teachers to teach mathematics by enacting RDM.

We view RDM as an approach that falls under the larger cat2015egory of ambitious mathematics teaching. According to Anthony and colleagues (), ambitious mathematics teaching “involves skilled ways of eliciting and responding to each and every student in the class so that they learn worthwhile mathematics and come to view themselves as competent mathematicians” (p. 46). RDM can afford opportunities to elicit students’ thinking by explicitly welcoming draft ideas or in-progress ideas, which can help students view themselves as competent in mathematics. RDM also can provide an avenue to respond to students’ thinking through workshopping and revising students’ drafts, which then can support learning mathematics through continuously improving students’ work and thinking. RDM has been found to have promise in an exploratory study in a high school in the Philippines; students who experienced opportunities to engage in RDM performed better on a measure of problem-solving skills compared to students who were not asked to engage in RDM (Felisida & Roble, 2020).

The purpose of this paper is to identify teachers’ enactments of RDM, as a case of ambitious mathematics teaching, and variations among these enactments. Ambitious teaching, such that “teachers teach in response to what students do as they engage in problem solving” (Kazemi et al., 2009, p. 11), is worth the effort if teachers want to develop their students’ mathematical proficiency (Kilpatrick et al., 2001), including fostering adaptive reasoning, conceptual understanding, and positive dispositions. This analysis contributes to building a knowledge base for ambitious mathematics teaching (c.f., Hiebert et al., 2002) by documenting practitioners’ work in a public, sharable manner through the voices of teachers. In this paper, we interpreted how variations among teachers’ descriptions of enactments of RDM revealed efforts to achieve different goals. The following research questions guided this study: When teachers described enacting rough draft math, which teaching practices were salient and feasible? Among these salient and feasible enactments of rough draft math, how did teachers’ descriptions vary and in what ways could these variations be viewed as potentially having productive and/or powerful impacts?

Situating rough draft math within ambitious mathematics teaching practices

In their review of mathematics teaching practices that support students’ learning, Hiebert and Grouws (2007) identified two impactful teaching practices: making concepts explicit and engaging students in opportunities to grapple and make sense of mathematics. This second practice, opportunities to grapple, has also been described as productive struggle, which occurs when students put forth effort to understand what is not immediately clear to them (Warshauer, 2015). The spirit of this approach to teaching is to position students as active sense makers as they communicate to learn (Sfard, 2001), or talk, write, and represent their thinking to continue learning. One strategy teachers have used to engage students in grappling with mathematics has been error analysis (Borasi, 1994). RDM is a related process that engages students in grappling as they share their emerging, in-progress ideas and have opportunities to analyze and learn from them; however, students can also revise correct explanations by enhancing them to become more elegant, precise, or illustrative.

The concept of a growth mindset (Dweck, 2006) has been argued to support students with grappling to make sense of mathematics (Boaler, 2022). Dweck’s (2006) theory of mindset indicates that people may view their ability as fixed, or unable to be changed, or from a growth mindset, such that they view their ability as malleable and capable of improving. Students who reported holding a growth mindset have been more persistent in the face of challenge and academically higher performing (Blackwell et al., 2007). Sun (2019) argued that to promote a growth mindset, it is not enough to communicate verbal messages to individual students to adjust their mindsets. Rather, systemic structures in schools, such as assessment practices, tracking structures, and teaching approaches, need to be aligned with a growth mindset. Effective mindset interventions do not only tell students about growth mindset, but they provide students with clear opportunities to put such a mindset into action (Yeager & Dweck, 2020). RDM could address this challenge of aligning instructional practices with intended messages about the learning process; inviting students to revise in mathematics allows students to continuously improve their thinking over time, which could put a growth mindset into action.

Another teaching approach used to support students with grappling to make sense of mathematics has been described as five key practices for mathematics classroom discussions: anticipating, monitoring, selecting, sequencing, and making connections between student responses (Stein et al., 2008). RDM can be woven into these five practices. Regarding drafting, teachers can anticipate students’ initial rough draft ideas, and monitor, select, and sequence drafts to be discussed. Connecting students’ strategies has been viewed as a challenging step in these five practices (Ayalon & Rubel, 2022). The process of workshopping students’ drafts can aid teachers with fostering these connections as students reflect upon what is similar and different between multiple students’ drafts, which can inspire revisions. Additionally, if teachers enact RDM by inviting students to reflect on how their thinking changed during a discussion, they could potentially add a sixth practice—student reflection—to these five practices.

Decomposing the teaching practices in rough draft math

To understand ambitious teaching, mathematics teacher educators have decomposed and specified teaching practices, which, according to Grossman (2011), centers the work of practitioners in the research process. Decomposing teaching practices “involves breaking down practice into its constituent parts for the purposes of teaching and learning” (Grossman et al., 2009, p. 2058). Teaching practice can be decomposed by examining a range of artifacts, such as written narratives, cases of practice, videos, lesson plans, student work, or live observations. This work can lead to developing a “grammar of practice” or “a specific technical language for describing the implicit grammar and for naming the parts” (Grossman et al., 2009, p. 2069).

Once categories for describing practice have been developed, these descriptions can be used in teacher education. Identifying discrete instructional moves “allows for targeted practice” (Grossman, 2011, p. 2940). If mathematics teacher educators name and label teaching practices by identifying specific instructional moves through analysis, both researchers and teachers are better able to identify ambitious teaching in action. Once teaching practices are named and specified, more teachers can intentionally plan to enact them, put them into practice, and reflect upon their efforts to enact them.

Efforts to decompose teaching can support the learning of in-service teachers. According to Sztajn and colleagues (2020), “when practicing teachers learn about a practice that is complex and new to their repertoire, no matter how long they have been teaching, it is appropriate to consider them as novices to that particular practice” (p. 3). Identifying the specific techniques of rough draft math can support more teachers with learning to enact rough draft math.

RDM, or incorporating rough drafts and revising into mathematics class, can involve some challenges for teachers. Teachers need to develop facility with eliciting and responding to students’ thinking (e.g., Fraivillig et al., 1999). The role of listening to understand students’ draft ideas, both teachers listening to students and students listening to one another (Hintz et al., 2018), is amplified when teachers enact RDM. Building on students’ thinking (Leatham et al., 2015) so that revising takes place is demanding in part because it can be challenging in-the-moment for teachers to recognize strengths (Jilk, 2016) in students’ draft thinking and to manage classroom interactions to position students’ contributions as valuable for their peers’ learning (Wood, 2013). If RDM teaching practices are decomposed and specified, they could be more accessible for teachers to enact.

Listening to teachers’ voices about rough draft math

Teachers’ voices about their practice can provide valuable insights about the work of teaching. Analyses of teacher voice can honor creativity in teaching and afford a situated understanding of professional decisions in local contexts (Atkinson & Rosiek, 2008). Teachers’ descriptions of their teaching can offer greater resonance and credibility for those who strive to enact similar work. Resonance and credibility with readers are two criteria for quality in qualitative research (Tracy, 2010).

We recognize complexities with reporting teachers’ voices. Following Atkinson and Rosiek (2008), we are aware that teachers’ descriptions of their teaching do not capture episodes of practice as a video camera would, and their self-reports of teaching are a meaning-making activity in themselves. However, listening to teachers’ voices can reveal what teachers intend to do when enacting RDM, including what is important (or salient) to them and what teachers identify as possible to enact (or feasible) in their classrooms.

Rough draft math in mathematics classrooms is an idea that has been developed through listening to teachers’ voices since 2016, and this study continues that effort. Our perspective aligns with Cai and colleagues (2017):

We believe research has a greater impact on practice when teachers play a purposeful role in the research process, whether in defining research problems; in identifying learning goals, subgoals, and learning opportunities; or of course in implementing learning opportunities in the classroom. (p. 466)

Listening to the voices of teachers about which teaching practices are more feasible to take up can help mathematics teacher educators support more teachers with enacting rough draft math.

Rough draft math was co-developed with secondary mathematics teachers in a study group in 2016, facilitated by the first author. Teachers participated in the study group because they wanted to co-develop strategies to improve classroom discourse so that more students felt safe to participate in discussions. In the study group, teachers read and discussed Exploring Talk in School (Mercer & Hodgkinson, 2008). Exploratory talk was described as hesitant, tentative, and rough draft in nature (Barnes, 1992), in contrast with talking to perform (final draft talk). As teachers generated ideas about how to promote exploratory talk, they decided to shift to use the phrase “rough draft talk,” following Barnes, because they conjectured that their students would better understand that phrase, compared to “exploratory talk.” These teachers created and shared strategies to welcome students to share their draft thinking. Next, these teachers recognized that drafts should be revised, so they developed strategies for integrating revising into mathematics teaching. The work of this teacher study group culminated in an article for teachers, co-written by some of the study group participants (Jansen et al., 2016). As the first author continued to study additional ways that teachers have enacted RDM, she wrote a book for practitioners to curate and share these approaches (Jansen, 2020a).

Salient and feasible enactments

This study was conducted to understand what teachers found salient and feasible when enacting RDM after texts for practitioners were published (Jansen et al., 2016; Jansen, 2020a). We viewed salience as reflecting the importance teachers placed on an instructional enactment. Salience has been described as a level of priority (Mitchell et al., 1997); high salience can reflect high importance. Mitchell and colleagues (1997) applied the concept of salience to identify who, or which group of stakeholders’ perspectives, had priority. In this study, we identified which enactment practices were salient for teachers. We viewed feasible enactments as those reported by many participants, because feasibility in teaching has been described as reflecting enactments that are available and accessible to put into practice in teachers’ local contexts (Sahakyan et al., 2018).

We wanted to investigate which enactments of RDM were described by teachers in ways that reflected a high level of importance for teachers (salient) and were described as accessible to enact in most participants’ contexts (feasible) after a minimal intervention. Reading a book could be viewed as a relatively minimal intervention to support teachers’ learning. It is worth investigating which enactments of teaching that teachers describe as salient and feasible after reading about an approach to ambitious teaching.

Identifying variations in ambitious teaching

There are likely to be variations in how teachers enact ambitious teaching practices. In their investigation of teachers’ enactments of the ambitious teaching practice of scientific argumentation, Berland and Reiser (2011) identified that argumentation entailed three goals: making sense of a phenomenon, articulating those understandings in an argument, and persuading others. If the primarily foregrounded goal was sensemaking, discourse could sound more like trying to understand one another and revising accordingly, while a primary goal of persuading could be reflected in discourse that could sound like defending or evaluating arguments (Berland & Reiser, 2011). One way to understand variations in teaching is identifying alignments between potential goals and enactments. In this study, we strive to understand teachers’ underlying rationales behind enactments of RDM.

While investigating teachers’ enactments of rough draft math, we strove to honor teachers’ professional decision making. Teachers formulate goals for their students and make decisions about what to enact and how to enact it (Penuel et al., 2014). We do not view enactments of RDM through a narrow lens; there is not one single way to enact rough draft math with integrity. Rather, RDM is a concept that teachers can use to consider how they can create learning experiences such that students share their emerging thinking, treat students’ thinking as having strengths to build upon, and provide students with opportunities to revise their thinking.

Powerful and productive variations

Variations in enactments of teaching can be both powerful and productive. Productive variations are those that have the potential to support students’ learning of mathematics, and powerful variations are those that have the potential to support the development of students’ positive identities toward mathematics (Herbel-Eisenmann et al., 2013, 2015). Understanding how enactments vary can reveal teachers’ differing goals for their enactments, such as their efforts to provide students with different productive and powerful learning experiences.

Methods

In this study, we took a phenomenological approach to studying enactments of rough draft math by exploring descriptions of lived enactments of RDM through listening to teachers’ voices. The purpose of phenomenological research is to understand a concept or phenomenon (in this case, RDM) through the lenses of other people (Creswell, 2007). In our analysis, we strove to understand how teachers described enacting rough draft math or how they experienced the phenomenon (Boadu, 2021). We present this evidence to illustrate how this population of teachers conceptualized their teaching practice (Harris, 2011). Through characterizing RDM through teachers’ voices, we hope that our findings provide credibility and resonance for researchers and teachers (Tracy, 2010). We aim to conceptualize the phenomenon of RDM understand variations in how this concept is enacted in the classroom.

Participants and context

We interviewed 32 mathematics teachers from eight states in the USA. We recruited teachers who had (1) participated in book study groups discussing Rough Draft Math: Revising to Learn (Jansen, 2020a), (2) read the book on their own, and/or (3) had participated in related professional development focused on RDM. Of the participants, 81.25% reported that they had read the book in the context of a book study or on their own. The rest of the participants (18.75%) reported participating in professional development about RDM, such as attending talks given by the first author. Most participants were recruited through recommendations by teacher leaders who had facilitated a book study (e.g., math coaches or district curriculum directors). We also recruited participants through social media.

Participants in our sample taught a variety of grades in K-12, from third grade through AP Calculus in grades 11 and 12. Participants reported a range of years of teaching experience from 2 to 35 years. Participants taught in eight different states in the USA: four states in the West (Arizona, California, Oregon, and Washington) [65.6%], two states in the Midwest (Illinois and Ohio) [18.8%], and two states in the Mid-Atlantic region (Delaware and Maryland) [18.8%]. Eighty-one percent of the participants self-identified as women and 19% identified as men. Seventy-five percent of participants self-identified as white, 9% as Asian, 9% as biracial, 3% as Latinx, and 3% as Jewish.

All researchers were former classroom teachers. The first author had been a middle school teacher, the second author had taught elementary school, and the third author had taught high school. They all share a value for teaching mathematics in ways that promote drafting and revising and that centers students’ reasoning.

Data collection

The first and third author conducted one-on-one interviews remotely with each participant over Zoom. Interviews lasted between 35 and 60 min and were semi-structured. Participants chose their own pseudonyms. Data for this study were collected in the 2021–2022 academic year.

Prior to the interview, participating teachers were asked to send a digital artifact via email that provided an example of how they incorporated RDM into their teaching practices. Artifacts were used in interviews to ground teachers’ descriptions of their enactments (Grossman, 2009). Examples of artifacts included samples of student work, handouts of mathematics tasks, links to online activities, slide decks used during lessons, videos of classroom events, assessments with directions that prompted drafting and revising, and photographs of students.

All participants were asked the same initial question and were then prompted to elicit further descriptions of their enactments of RDM. Interview questions included the following: If you were to tell a colleague about rough draft math, how would you explain it to them? Tell me the story of this artifact. Why did you decide to do this in your classroom? How is this artifact an example of rough draft math? Is there anything else related to “rough draft math” and revising that you’re doing with your students in class?

Data analysis

We conducted data analysis in multiple stages. Initially, the first and third author wrote analytic memos (Saldaña, 2013) to document researchers’ initial drafts about how teachers described enactments. We defined an enactment as an action that teachers described putting into practice. Informed by the analytic memos, we inductively developed descriptive codes (DeCuir-Gunby et al., 2011; Saldaña, 2013) of names of enactments.

Coding was an iterative process. First, the first and third author coded approximately one-third of the interviews to develop and revise the codes. We initially developed codes for enactments in parallel with the themes in the book that teachers read: culture (building and sustaining a culture to welcome students’ drafts), using tasks (selecting, modifying, and implementing tasks to invite drafts [multiple explanations and/or multiple strategies]), and revising (explicitly inviting students to revise). Additional codes for enactments emerged inductively (Strauss & Corbin, 1998) from the teachers’ interviews, such as collaboration (students workshop drafts in partners or small groups). Once we developed a stable coding scheme, the second and first author applied the codes to all interviews for the first level of analysis. Every interview was coded by two researchers, and they met to resolve disagreements.

Salient and feasible enactments

For our second level of analysis, we interpreted which enactments were salient for each teacher and feasible among the sample of participants. We determined an enactment to be salient for a teacher if she or he mentioned the enactment frequently (more than twice across an interview) and in detail (more than two lines of transcript text), in contrast with enactments mentioned briefly and only once. Thus, we assumed that if a teacher talked about the enactment repeatedly and spoke about it with elaboration, that enactment was important (or salient) to that teacher. We identified feasible enactments by determining which salient enactments were described by most of the participants. Thus, we assumed the enactments that were most feasible to enact were the salient enactments that most participants reported putting into practice. The first and second author conducted this coding and met to ensure agreement.

Powerful and productive variations

For our third level of analysis, we identified variations within enactments. After we identified the most feasible and salient enactments, we compared and contrasted ways that teachers described their teaching within each of these enactments. To interpret how variations in enactments could be potentially productive, we interpreted what teachers’ descriptions appeared to reveal about their efforts to support students’ learning of mathematics. To interpret how variations in enactments could be potentially powerful, we interpreted what teachers’ descriptions appeared to reveal about their efforts to develop students’ positive dispositions or identities in mathematics.

Results

Our first research question was: When teachers described enacting rough draft math, which teaching practices were salient and feasible? Two enactments were described by most of these teachers (78.5%) with salience. Thus, two salient enactments appeared to be the most feasible: (1) incorporating revising into mathematics learning and (2) selecting and implementing mathematics tasks to invite drafting and revising.

Our second research question was: Among these salient and feasible enactments of rough draft math, how did teachers’ descriptions vary and in what ways could these variations be viewed as productive and/or powerful? Regarding variations in teachers’ descriptions of incorporating revising, (1) teachers described providing students with both structured and unstructured revising opportunities during mathematics lessons, and (2) teachers described providing students with opportunities to revise assessments in different ways, such as revising their work on a single assessment (e.g., test corrections) or reflecting on how their thinking changed across a set of their work (e.g., self-assessment portfolios). Regarding variations in teachers’ selection and implementation of mathematics tasks, (1) teachers described selecting different types of tasks to invite drafts and revising (tasks modified from their curriculum materials and instructional routines) and (2) teachers described implementing tasks with revising either to reinforce students’ learning of previously taught content or to support students with developing understandings of new content. Below, we describe these variations in greater detail, through teachers’ voices, and we interpret how these variations could be viewed as powerful or productive.

Variations in enacting revising

Revising was one of two enactments described with salience (repeatedly and in detail) by 78.5% of the participants (25 out of 32 teachers). Because such a high number of teachers reported this enactment with salience, incorporating revising appeared to be a feasible enactment for these participants. Teachers described revising as having students improve work they had already drafted (revise their work). Additionally, some teachers described inviting students to reflect on how their thinking changed after doing some mathematics (revise their thinking).

Structured revision opportunities

Teachers described providing structured and unstructured opportunities for students to revise. First, we will give an example of a structured revision opportunity from Ms. Parra’s description of how she enacted revising. Ms. Parra self-identified as a Hispanic woman, taught sixth grade in the West, and had eight years of teaching experience. To structure students’ revising, she asked students to make space to document their revised thinking. She had students draw a line down the page to preserve space for a first draft and then a successive draft.

I told them to just put a line, to fold their paper and put their first thinking and then their next thinking. They felt okay with making the first thinking messy. … if they labeled it first thinking, then they were okay with everything being messed up and not erasing it.

Giving students a designated location on a piece of paper with room to record an initial draft and space to document changes in their thinking provided a structure for revising (Fig. 1).

Fig. 1
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Structuring revisions

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This structure explicitly clarified that students would have more than one chance to draft and revise their thinking. She reported that labeling their work as “first thinking” or rough drafts led to more of her students being willing to write down their thoughts.

Ms. Parra’s rationale for giving students structured revising opportunities aligned with how she defined rough draft math. When asked, she defined RDM as follows:

I think that we all have this idea of rough draft being language arts and having like lots of messy copies with lots of writing all over it and edits and things. In math, we never have all these multiple sheets of the same problem that we've been thinking about for an extended period of time. And why not? To me, math is an ongoing thing.

Ms. Parra described a way to put this process into action, because she asked students to document their evolving thinking. Structured revising opportunities could help students see how their thinking is shifting and potentially improving.

Unstructured revision opportunities

Sometimes teachers described inviting students to revise in less structured ways, at any time, not in response to prompts. Ms. Dougherty encouraged her students to revise continuously, as a regular practice, even without an explicit structure during a classroom activity. She taught seventh grade in the Mid-Atlantic region of the USA, identified as a white woman, and had twelve years of teaching experience. Ms. Dougherty gave her students an open invitation to revise at any time, and she reinforced that changing one’s thinking is a strength that should be preserved, not erased and hidden:

If they want to add anything, and usually if I have them add thinking or change thinking, I ask them not to erase, but to use a different color. And just kind of add any thinking or, even if another peer of theirs is talking to them, they can even tell me who told them that information… Anything they want to, really, as long as they’re getting that support to understand. We just keep that in mind, that it’s not about the answer as much as the understanding.

Using another color and avoiding erasing could afford opportunities for students to see the evolution of their own thinking and validate that it is normal to continuously improve. She also wanted to encourage her students to see that a purpose of learning mathematics was to continue to understand, not just get a correct answer. Revising is less structured in this example because she said revising was available to students at any time, in contrast with explicitly providing students with space to write more than one draft.

Mrs. Dougherty described RDM as learning by revising and building upon your initial thinking:

It's simply your first draft of thinking … You haven't revised any of your thinking yet. It's just your first draft… from there we build on that, and we use our rough draft to help build that understanding to make it stronger and understand where we came from along the way.

She encouraged her students to take initiative and use any opportunity they saw to keep a record of how their thinking evolved, although not always through a structured revision opportunity.

We conjecture that both structured and unstructured opportunities to revise in mathematics are enacted by teachers for a similar purpose: Students are given multiple attempts on a task and document changes in their thinking. This purpose suggests ways that these enactments could be both productive and powerful for students. It could be productive because revising is a part of learning. Additionally, if students have opportunities to document their revisions, this could be productive because such documentation serves as a record of their learning over time.

We also conjecture that revising opportunities can be powerful because they can support students with developing positive identities as mathematics learners. If students see that their thinking can improve, it is possible that they could develop a growth mindset (Dweck, 2006; Sun, 2018). Additionally, if students see that their initial draft thinking has assets that can be built upon, they may develop a stronger sense of competence as mathematics learners. Also, as peers recognize how and what they learn from one another, they can experience validation from how they help others and can come to value the thinking of peers in their classroom community. See Table 1 for a summary of potentially productive and powerful variations of enactments of RDM.

Table 1 Variations in enactments of rough draft math
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Revising in assessment: test corrections

Some teachers incorporated revising into their assessment practices, and they did so in two different ways: by inviting students to correct mistakes on tests or by engaging students in self-assessment. Ms. Heeler, a high school teacher in the Mid-Atlantic who identified as a white woman and had seven years of teaching experience, provided her students with opportunities to keep learning through correcting their tests:

I have seen students that normally wouldn’t try very hard in class going back and revising their tests, bringing their grades up. Students have communicated to me, I used to not like math, but Ms. Heeler made me like math again.

This practice of revising tests aligned with Mr. Heeler’s definition for RDM, which she described as follows:

…sharing unfinished ideas and incomplete thoughts, making a rough draft or prediction, and then going back and evaluating it and making changes and reflecting… opening the door for everyone to contribute because even a rough, unfinished thought has value that contributes to the rest of the class.

Test corrections could be viewed as productive in supporting students’ learning, because students would have opportunities to improve their work, by correcting mistakes, which can improve their thinking. Ms. Heeler also described powerful benefits of test corrections, such as students’ improved effort, better attitude toward mathematics, and increased recognition that their thinking can help others learn, which suggests that test corrections specifically (and rough draft math generally) could potentially positively impact students’ identities.

Revising in assessment: student self-assessment portfolios

Another way that teachers connected RDM to assessment was through incorporating student self-assessment. Mr. Vandelay identified as a white man who taught high school in the West in the USA, and he had twenty-four years of teaching experience. He said:

I’m using an ungrading model in my classes this year. …there’s just a lot of issues involved with using traditional tests as, as artifacts of evidence of student understanding… I just decided that those [tests] can become pieces of evidence, but they shouldn’t be the basis of evaluation. And so I’m using a portfolio as the basis of evaluation this year, and it’s very much student-curated and student-led, all the way down to self-evaluation… the ethos of rough draft math is always there because until the final grade report, everything is formative.

His use of “ungrading” (Blum, 2020) referred to the student self-assessment process, because students were not subject to the grades assigned to them by the teacher. Tests were considered one piece of evidence of student work, among multiple, that could be used by students to reflect upon how their thinking has developed when they created their portfolios. Mr. Vandelay went on to say:

… it was a way for me to satisfy this years’ long struggle with not having progress and growth… It just feels very human and organic. It almost feels like there’s no grading at all. Grade reporting just becomes a task, but grading… it’s not something my students are asking about anymore. So, it just feels like learning, which is what I think is the core of rough draft… it’s removing student thinking away from the stress and the typical experience of I have to do well on this to get good grades. Its intent is to look at assessment and evaluation in the lens of always progressing, always growing.

He provided this structure of self-assessment to communicate to students that our thinking is always growing and changing, which he reported as “the core” of RDM.

Student self-assessment could be viewed as a productive and powerful variation of RDM. Self-assessment seems to be potentially productive in that, according to Mr. Vandelay, “it just feels like learning.” This could mean that through self-assessment, learning continues as students develop new understandings by looking across their work. Additionally, Mr. Vandelay’s description suggests that self-assessment could be powerful in that students can develop autonomy as teachers transfer authority for determining grades and identifying what students understand onto their students. Assessment is not only about documenting performance, but it is also about analyzing for growth, which, again, could promote a growth mindset. Mr. Vandelay also observed that self-assessment could possibly reduce the stress for students that is typically associated with assessment in mathematics.

Variations in task selection and implementation

Teachers’ use of mathematical tasks was the other enactment described with salience (repeatedly and in detail) by 78.5% of the participants (25 out of 32 teachers). Because such a high number of teachers reported this enactment with salience, task use was also a feasible enactment for these participants. Teachers described selecting mathematical tasks to allow for more possible drafts to be shared, either through modifying a task from their curriculum materials or enacting an instructional routine with intentionality. Additionally, teachers described implementing tasks with revising to either reinforce previously taught content or to develop understandings of new content.

Selecting tasks to welcome more drafts: modifying curricular tasks

Teachers described how task selection was a part of their enactment of RDM, particularly the role of the task in eliciting a wide range of students’ rough draft thinking. Ms. Alderman, who identified as a white woman with eight years of teaching experience and who taught sixth grade in the West, spoke about needing to get past the hurdle of finding special tasks for enacting RDM. She realized she could use the tasks in her own curriculum materials. She shared an artifact of a task from her textbook that she and her co-teacher modified to invite rough drafting and revising. The original task asked students to write and evaluate an expression to find how many pounds of plants a herd of 18 elk can eat in one week, and the book provided information that an elk can eat 20 pounds of plants each day. She modified the task so that it read as follows:

In an ecosystem like the rainforests of Washington, some animals get energy from eating plants. An elk eats an average of about 3 pounds of food for every 100 pounds it weighs! A typical elk eats about 20-24 pounds of plants in a day!

When describing the task modification, she said.

I know that for me, I got really hung up on that question like, well, what is a task? Where can I go to get those tasks? …okay, so it’s more about this process of like, what is happening here? I think the biggest thing is -- there’s no question there. We took out the question. We made it relevant, and with that notice and wonder, without a question there.

Additionally, Ms. Alderman enacted the task by including a photograph of an elk taken by her teaching partner, Mx. Neil, while on a hike locally in their state. They enacted this task by having students share what they noticed and wondered. Based on the students’ wonderings, the class looked up more information to address some of the wonderings, such as how much an elk might weigh. To address additional questions posed by students, the class wrote a variety of mathematical expressions and evaluated them, including the amount of food that a herd of elk (in this case, 18 elk in a herd) would eat for ten weeks and then any number of weeks. The modifications to the task not only connected to students’ local setting, but more than one correct expression was possible, which could lead to more possible drafts and more opportunities to revise.

Modifying this task had the potential to lead to productive conversations that supported students’ learning. When writing the expressions as a class, Ms. Alderman reported that students strove to understand each other’s expressions and revised their own expressions. Ms. Alderman said,

[that she was]…really able to then push the students who are struggling to understand where the pieces [in the expression] go. They asked really deep, meaningful questions. Like why, why did you put that there? What does that mean?”

She reported that she “wanted [them] to have meaning behind what they were doing.” When Ms. Alderman described RDM, she spoke aloud about what she might say to a colleague: “Why can’t we revise math? Why does it have to be this right and final answer?” She appeared to align opportunities to revise with opportunities to continue learning.

She also described how this task modification could be powerful and support the development of students’ identities. She said that the modified task could reduce “the pressure of being correct” because “there wasn’t a question. And so it was like, ‘well, I can’t be right. I can’t be finished. Because there’s not a question.’” She found that this task revision appeared to support students with persisting to understand the meanings underlying the symbols in the expressions through drafting and revising.

Selecting tasks to welcome drafts: using instructional routines intentionally

Teachers also described RDM as providing inspiration for ways to more intentionally enact instructional routines that they already implemented. They noticed connections between the structure of the routine and the processes of drafting and revising. Ms. Zafar identified as an Asian woman with four years of teaching experience, and she taught fourth grade in the Midwest. She reported launching her mathematics lessons with an instructional routine related to counting. Students were shown an image and were asked to determine what unit they wanted to count and share their process of counting. [How many? (e.g., Danielson, 2018) What did you count? How did you count them?] Ms. Zafar used the title Contemplate then Calculate for this routine with her students, which is the name for this routine used by Kelemanik and colleagues (2016).

For her artifact, Ms. Zafar shared an image of candy that she found on a website (see Fig. 2).

Fig. 2
figure 2

Contemplate then calculate (how many?) image from Ms. Zafar’s task

Full size image

The image (Acosta, n.d.) consisted of six groups of three candies in each group, and each group of candies was a different color (e.g., a group of three red candies, a group of three green candies, etc.). Not only did students describe how they saw eighteen candies as six groups of three candies, they also shared different ideas and relationships. Students reported that one-sixth of the candies were green or three out of eighteen candies were green. Some students identified one inner circle of candies and one outer circle of candies.

Ms. Zafar reported that RDM helped her move away from expecting one correct answer from students when enacting these routines toward becoming more open to a variety of ways of seeing mathematics in the image. She said:

Initially, before rough draft math… students would say something like, there’s three orange M&Ms, and three brown M&Ms, and three yellow ones. And so they’d give me a total number. ‘I think there’s like six groups,’ they would say. ‘Oh, there’s 18 M&Ms.’ So that almost feels like a final answer, right? …How do we build on just that piece?

Before she taught with RDM, she found less to discuss, as the discussion narrowed onto a single solution. After reading about RDM, she encouraged a wider range of draft ideas:

[With RDM,] I would say, okay, tell me what the numbers mean or share your thinking with us… so we talk about fractions, we can talk about multiplication, adding as repeated addition… we had our geometry unit… M&Ms are circles and a circle is 360 degrees.. one-sixth are orange, one-sixth are green, making like a little pie chart…

Ms. Zafar selected this instructional routine to use to promote RDM because she saw that it would allow the class to be open to a wide range of students’ draft ideas. Many different perspectives could create more opportunities to revise by expanding one’s initial thinking.

Ms. Zafar’s use of this routine connects with the definition of RDM that she reported in her interview:

Where we’ve heard the term rough draft is really with writing…I would refer to that initially to say, well, that’s just the rough draft, it’s just the first copy…And we know we have to revisit it. We know we have to edit it and go back to it. So I love that idea of when it is revising to learn, but building on what you’ve already thought and doing it differently. So when we think about a rough draft of something, it’s not the final copy, but also it’s not wrong, even though it’s not finished yet.

She incorporated this instructional routine to provide a structure that illustrated to students that all of their draft ideas were welcome and had merit.

She viewed this instructional routine as productive because she connected it with learning experiences for students. Ms. Zafar said that, during this routine, students were “learning from each other as well, which goes back to that revising aspect of rough draft math, where you are learning from each other, where you're open to thinking about it in a different way.” If students are open to seeing differently, they can revise (and learn) by expanding their perspective.

Ms. Zafar also described that this routine was powerful for promoting students’ positive mathematical identities, because so many students shared their thinking:

This was a very low risk, open playing field where they were raising their hand, if nothing else that they would raise their hand for, with our whole group lesson, this was the time that they would raise their hand and share their thinking and participate. It gave them a voice.

She engaged students in a routine with many ways to share valid draft ideas. She then said, “they know that this [her classroom] is a place where that [their idea] is welcome and celebrated… I want them [her students] to believe in themselves.” She wanted her students to have experiences of being validated for sharing what made sense to them in mathematics. She valued this routine because the images represent objects that students could encounter outside of school. She said, “I think what makes learning relevant and interesting, engaging, is when they see the value of it in everyday life, um, that they see it everywhere and like, okay, yeah, we can actually use this.” Students could learn to see mathematics as a part of their lives.

This was not the only instructional routine that teachers described using to enact RDM. Ms. Alderman incorporated a Notice and Wonder routine (Rumack & Huinker, 2019) when she engaged her students in her modified curricular task. Some teachers also described inviting students to draft and revise as a modified approach of a routine called Stronger and Clearer Each Time (Kane, 2019; Zwiers et al., 2017), which involves students writing down their first thinking, then partners share their first thinking with each other, and students write down their revised thinking as informed by learning from a partner. Ms. Hunter, a teacher in the West, said, “I like the stronger and clearer… they can just think their initial thinking at the beginning of a unit and see how that grows, how they can learn from each other, and that thinking grows.” Teachers noticed that instructional routines that they already used could connect directly with the idea of RDM.

Implementing tasks with revising: reinforce content

Two different ways that teachers implemented tasks when enacting RDM was encapsulated by this quote from Ms. Green: “So, are we working solidifying an idea? Are we working on learning a new idea for the first time?” She was a high school mathematics teacher with sixteen years of experience who taught in the West and identified as a white woman. Solidifying an idea aligns with the idea of reinforcing content that was previously introduced.

Ms. Green enacted RDM to reinforce previously taught content by identifying students’ mistakes, while they were working on mathematics during class, taking a photograph of students’ work, and shifting to a whole class discussion to analyze the work. She said:

I love to use rough draft math to highlight really good mistakes as well. And so this is where the student’s thinking is right now. What mistake were they making and how do we move them to that revision?... why was it so good and how can we revise it to make it the accurate answer, that final draft thinking?

This practice reflects another instructional routine: My Favorite No (Caniglia, 2020). This routine is a structure for error reflection and analysis. The teacher implements a task in which students are asked to calculate a solution; then, the teacher collects and analyzes the solutions to identify a common (“favorite”) mistake that exemplifies a way of thinking that the teacher wants the class to analyze. The teacher presents the solution anonymously. The students identify what is wrong with the response, reflect upon why the student thought that way (to honor the thinking process), and correct the mistake. Other teachers have referred to this practice as “my favorite rough draft” (Jansen et al., 2016).

Ms. Green said that the idea of RDM impacted her teaching because she changed her teaching to incorporate students’ thinking when it was not completely accurate or complete, validate the thinking, and analyze the thinking in discussion as a learning opportunity for her students:

Probably the biggest change is that I used to wait and find final drafts… I waited until I could find a final draft wrong answer, or a final draft right answer, or a final draft almost-there answer. And this idea that you can pull student thought at any moment, at any time span within doing the mathematical task, at any given moment, if there’s a great opportunity to snag work and orient the rest of the class to it, because there’s some good mathematical or teaching reason to do that, that timing didn’t matter anymore. It wasn’t about waiting until the student’s work was complete, which I felt like was an old habit of mine, but there’s so many different places to stop and snag the work that is gonna help the class as a whole.

By recognizing that students could learn from thinking at any stage of the drafting process, Ms. Green changed her criteria for how she would select student work for discussion. This shift afforded her with more flexible timing during teaching without having to wait for students to complete their work before discussing it.

Ms. Green’s approach aligned with both potentially productive and powerful outcomes. Her enactments could be viewed as product in her efforts to support students with reaching a “final draft,” or a correct and complete solution. Powerful learning aligned with sharing authority with students to analyze the work in discussion, positioning students as capable of making sense of mathematics. Ms. Green remarked on this point, stating “…it really has changed the mathematical authority in the room.” She also said that her students became stronger risk takers, as she said, “…some of the things I really love about it is that students aren’t scared to try. They’ll always just get started…” after experiencing RDM in this way.

Multiple high school teachers described using RDM to solidify ideas to prepare for AP tests (AP calculus, AP statistics). Ms. Apple, also a white, female high school teacher with 16 years of experience in the West who taught AP Calculus said:

All the tests are just kind of a checkpoint just to see where we are. I think of the AP exam as like the final draft, right in line that we want to get to. So all along the way, I want them [students] to continue to think of each unit assessment as, as a chance to revise their thinking and get feedback on where they are and what they need to do to be ready.

For these teachers, getting students to a “final draft” was an important goal due to preparing students for a high stakes assessment at the end of the school year. However, students can keep making new connections and extensions of their thinking, even when they are able to draft a correct and complete solution.

Implementing tasks with revising: develop new understandings

Teachers also described their use of RDM to introduce a new idea. Ms. Shisler, who identified as a white woman teaching seventh grade in the mid-Atlantic with 24 years of teaching experience, reported using RDM in relation to what she described as modes (mode one: independent thinking time; mode two: collaborate with peers in partners or small groups; mode three: whole class discussion):

So I think all of my modes in instruction really feed in nicely with rough draft math because that first round, when they’re on their own, mode one, for that independent work and that processing time, that’s their initial thoughts and their ideas. And then when they get to collaborate in mode two with each other, that’s when they can start to hear ideas from each other and fix up their own ideas. And then mode three, when we share out as a class, they can, again, polish up their thoughts after seeing and hearing what the class thinks.

She was already using these modes of instruction, so RDM gave these modes particular purposes.

She asked her students to generate claims about what they believed could be true about relationships in mathematics and then work as a class to refute or affirm these claims. The claims they developed reflected new ideas that they were learning. She said, “…my artifact dealt with having students explore addition and subtraction of integers, and through that exploration period, they made some claims, because I really use rough draft math when students are exploring…” To develop their understandings of integer operations, she engaged her students in exploring patterns and asked students to draft claims about what they thought was true. Her artifact was a set of Google slides that displayed claims of students’ emerging or in progress understandings with the class. Ms. Shisler said:

We started to kind of develop a rule. So, after we went over as a class, I anonymized the kids’ [claims] and I showed some different ones, and they had to pick which ones they thought were correct and defend it.

Some of the students’ claims were as follows: “I think that we should make a rule about adding and subtracting, that subtracting an integer is the same as adding its opposite integer.” “I think subtracting a negative from a positive number makes the number bigger should be a rule, because, for example, 6 − (− 1) = 7. It’s like adding 1 to 6.” Then, after discussing the claims, students would engage in additional exploration and then have opportunities to revise the claims.

The use of RDM to introduce new content can be both powerful and productive. In Ms. Shisler’s classroom, this process has the potential to be productive because students can learn new content through generating and revising claims. Also, Ms. Shisler wanted the process of generating and revising claims to be powerful for students. She said, “…it [this claim work] builds their self-confidence.” She reported that RDM supported her effort to support positive identity development in students, as she said:

I started with the idea of promoting student agency and then I guess, you know, formalizing it more once I became familiar with [first author’s] idea of rough draft math. …every year I always want the kids to take ownership of their own learning. And I want them to ask those higher-level questions of themselves without me probing them. And I want them to challenge each other without me asking them…

Teachers like Ms. Shisler found RDM to help them reach goals that they valued, such as promoting agency while supporting students’ learning.

Discussion

These findings revealed two salient and feasible teaching enactments for rough draft math: revising and task use. It is possible that other enactments relevant to RDM, such as inviting collaboration among students and building the classroom culture to welcome students’ draft ideas, are practices that are valuable for ambitious mathematics teaching in general rather than specific to RDM. To elicit students’ thinking, which is a part of ambitious mathematics teaching (Anthony et al., 2015; Kazemi et al., 2009), it is helpful to intentionally foster a classroom culture that welcomes students’ thinking at any stage. To teach mathematics in ways that center students’ thinking, inviting students to collaborate with peers can be viewed as a part of that work generally. Revising in mathematics is specific to RDM, however, and variations in task implementation (reinforce content and develop new content) also aligned with revising.

Salient and feasible enactments

Revision and use of tasks were the most feasible enactments, as determined by the most teachers describing these enactments among enactments described with salience. Listening to teachers’ voices provided insights about salience of enactments. If we had only observed teachers, we would not have developed insights about which enactments were salient to teachers. We could have used an observation protocol to identify which enactments were feasible, but our observations would not have revealed which enactments were important to teachers.

Teachers described additional enactments, but they appeared to be less feasible among teachers in this sample. Other salient enactments included building the classroom culture to welcome rough draft thinking and providing students with opportunities to collaborate. However, revising and teachers’ uses of math tasks in RDM were the two salient enactments that were most feasible, so we focused our analysis of variations of enactments of RDM on these two enactments.

Powerful and productive enactments

We identified conjectures about how variations in enactments of RDM could align with different productive and powerful outcomes for students. Our lenses of productive and powerful variations suggest ways that RDM could be viewed as an equitable teaching practice. Two lenses on equity include access and identity (Gutiérrez, 2012). Providing students with access to learning opportunities is how we viewed a productive variation. Supporting the development of positive identities toward mathematics is how we viewed a powerful variation.

Variations in enactments appeared to align with potentially different goals or outcomes. Reflecting upon which outcomes could align with a variation in enactment affords teachers with opportunities to make more intentional choices about how they put RDM into practice with their students. For instance, if teachers intend to promote autonomy among their students as one of the potentially powerful outcomes, some of these enactments appear to be more closely aligned with that outcome, such as implementing tasks to develop and discover new content or incorporating revising into assessment through self-assessment portfolios. However, future research is needed to examine whether (and the degree to which) students experience these powerful and productive outcomes for various enactments.

Implications for mathematics teacher educators

These results suggest that it is possible for teachers to find it feasible to enact RDM, as a type of ambitious mathematics teaching, with relatively minimal support. These teachers enacted RDM after a relatively minimal intervention: reading a book (or, in a small subset of cases, participating in workshops about ideas from a book). None of these teachers had direct support from researchers or coaches about how to enact RDM beyond their book study or workshop participation. It is worthwhile to investigate through future research how much further teachers could go enacting RDM with additional support, or which teachers need what sorts of supports with what enactments of RDM. But it is promising that these teachers could enact RDM without extensive support.

Descriptions of enactments of RDM at a level of specificity, including how they vary and the potential outcomes they afford, provide insights for teacher education research and practice. These results illustrate options for enacting RDM that have been feasible for teachers to take up in their local contexts in a range of states in the USA, which suggests the possibility of other teachers being able to enact them. Future work could examine the feasibility of these enactments in additional contexts.

Additionally, these results could be useful for teachers to use as lenses to reflect upon. Teachers could consider which enactments they tend to use and why, which they do not, and which enactments they might want to attempt and what they might afford their students. Table 1 could be incorporated a tool to anchor teachers’ reflections as they contrast their purposes and desired impacts with various enactment approaches. Future research could examine how teachers use the results as lenses for reflection and for revising their own teaching practice toward more ambitious mathematics teaching. Thus, another way to think about incorporating revising into mathematics teaching is to revise and improve teaching practice.

Limitations

We conducted this study through teachers’ self-reports. Above, we reflected upon how self-report through teachers’ voices provided us with an opportunity to learn about the salience of enactments. However, it would be useful to observe their teaching to examine ways in which enactments align with and diverge from self-reports in future research.

We also only listened to the voices of teachers who volunteered to participate. Thus, we did not learn from the voices of teachers who may resist enacting RDM. Future work could attend to the voices of teachers who were required to participate in professional learning about rough draft math, not only volunteers.

Conclusions

This study afforded opportunities to identify enactments of ambitious mathematics teaching that were salient and feasible for teachers after they read about (or attended a professional learning workshop about) RDM. Incorporating revising into lessons and implementing mathematics tasks to invite drafting and revising were the two enactments that were described with salience by most of these participants. We documented variations in enacting revising, such as whether revising was structured or unstructured and ways in which revising was incorporated into assessments. We also documented variations in task selection and implementation, from modifying curricular tasks and using instructional routines intentionally as well as enacting tasks to either reinforce content or develop new understandings. These teachers’ voices contributed to building a knowledge base for teaching (Hiebert et al., 2002) about how teachers enact RDM.