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Types of and Purposes for Elementary Mathematical Writing Task Force Recommendations

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Teachers have long been encouraged to incorporate communication as a crucial component of learning mathematics. Writing in mathematics can support students’ learning of content and can provide students with another way to communicate their mathematical thinking. However, descriptions about the types of and purposes for mathematical writing fall short for teachers to carry out adequate instruction. Specifically, for elementary students starting in kindergarten, what types of writing can be leveraged to facilitate their learning of mathematics?

To address the lack of clarity about the types of and purposes for elementary mathematical writing, the Elementary Mathematical Writing Task Force was convened in October 2015. Members of the task force represented the fields of mathematics education, mathematics, and writing education; had expertise across the elementary grades; were knowledgeable about particular student groups, including English language learners, students who have learning difficulties, and/or those students identified as gifted; and had experience teaching writing and authoring assessments and curricula that include mathematical writing to ensure the group’s recommendations would be informed by and attend to various stakeholder groups and student needs.

The Elementary Mathematical Writing Task Force identified two overarching goals for mathematical writing: for students to reason and to communicate. The four types of mathematical writing recommended by the task force are: exploratory, informative/explanatory, argumentative, and mathematically creative writing. Each type of mathematical writing has unique purposes. The overarching goals, types of, and purposes for mathematical writing have the potential to provide students with an authentic way to engage with mathematics.

## Tutita Casa

Co-PresenterHow might you engage your students in the four types of elementary mathematical writing and their various purposes?

## Miriam Gates

A very interesting report. It seems to me that there are strong links to the Common Core Standards for Mathematical Practice (SMP) in the purposes of mathematical writing (and clearly across the entire video). I’m wondering if any work on the connection between these recommendations and the SMP has been done. If so, can you discuss a few of the connections as you see them?

## Tutita Casa

Co-PresenterGreat question! In fact, we relied on several standards documents to inform these recommendations, and the Common Core’s SMP’s were one of them. NCTM’s 1991 Professional Standards that explicitly attended to discourse and their 2000 Communication Standards did as well. None of these contained details specifically pertinent to writing, however. One document that provided more guidance for this work was the CCSS’s 2012 Framework for English Language Proficiency Development Standards Corresponding to the Common Core State Standards and the Next Generation Science Standards that attend to “productive language functions,” which includes communicating in writing.

## E Paul Goldenberg

I watched the video, downloaded the “Types and Purposes…” pdf and read parts (and skimmed all). I enjoyed both a lot, and found the content of the pdf both thoughtful and useful. What I was looking for (and did /not/ find, but perhaps because I was not careful enough) were comments about the limitations of writing or the limits one might put on its use. For example, does the cognitive load that writing presents (especially for the youngest students) ever compete with, rather than support, students’ attention to the mathematical content or reasoning? If so, how might teachers decide when to use writing and when not to?

## Janine Firmender

Co-PresenterGreat question! This would certainly be an important consideration when planning the tasks and approaches to engage students in writing about mathematics. While the current work was focused on the types and purposes of writing about mathematics that could be leveraged to develop and uncover students’ understanding, one way I see to support students’ attention to the mathematical content and reasoning would be to engage them in oral discourse prior to asking them to write about their ideas and helping them make connections between oral and written discourse. Future work could/should focus on the implementation strategies and considerations.

## E Paul Goldenberg

Thanks. I’d certainly love to hear more as you learn more. I’ve looked, though not intensely, to see what’s already known about the potential “competition” between attention to mathematical ideas and communication about mathematical ideas (orally or in writing) and have found nothing. It seems likely that something about timing matters—communicating our thinking with others clearly does help clarify our thinking in some cases, and writing our thoughts often lets us organize and clarify them, but these cognitively demanding tasks also take up mental time and space and attention. Do you know of any studies (or even theory) here? Again, thanks!

## Tutita Casa

Co-PresenterI agree that this is an important consideration for next steps in this work as teachers implement these types of writing. We have developed and field tested curriculum for k-5 students (Project M^3 for grades 3-5, M^2 for k-2) where, on average, students wrote about every 3-4 days. Anecdotally, this seemed to strike a nice balance and potentially provided students the space and time to solidify their understanding of key mathematical concepts through writing. You might find Cohen et al.‘s 2015 article in School Science and Mathematics discussing characteristics of second grader’s writing (M^2 intervention and comparison group) to be of interest. Both high- and low-performing students in the intervention group outperformed their counterparts in the comparison group with respect to sharing their reasoning as well as using precise mathematical vocabulary. While this study did not directly attend to your question about timing, it may suggest that the frequency of writing is a reasonable guideline.

## E Paul Goldenberg

Ah, thanks. This is, in fact, very helpful. Mostly, in classrooms, I see less focus on discussing reasoning (generally, less student talk) than seems sensible, and very little student writing, but where I’ve seen writing used with any regularity, it has been a daily required activity and not clearly purposed. The work you’ve put up dealt extremely well, I thought, with the purpose and nature of the writing, and now you’ve also clarified at least initial thoughts about timing. Nice!

## Tutita Casa

Co-PresenterThank you! We are hoping that the mathematical writing gets implemented with intention and not for the sake of writing in and of itself. We see writing as a tool that has the potential to further students’ reasoning, much like intentional implementation oral discourse and use of manipulatives, technology, etc. can when used in ways that support a mathematical habit of mind.

## E Paul Goldenberg

All the best to you. This is great.

## Jorge Solis

Thank you for sharing your work here! There are real synergies between literacy and content development. I was wondering what the recommendations say about addressing varied language proficiencies for ELLs in mainstream and/or bilingual programs?

## Tutita Casa

Co-PresenterThis is the first step we have taken to more clearly define what “mathematical writing” can entail for all elementary students participating in the regular mathematics curriculum. Attending to specific needs of student groups is an important follow up. We considered this long-term goal when identifying potential task force participants, and we have researchers and practitioners represented in the group representing the field of English language learners as well as special and gifted education. The task force also relied on the CCSS’s 2012 Framework for English Language Proficiency Development Standards Corresponding to the Common Core State Standards and the Next Generation Science Standards that you may already know addresses writing in mathematics.

## Courtney Arthur

This work shares some interesting findings. I am wondering how the task force was chosen and what recommendations and/or scaffolds are recommended for students who are far below grade level or struggle with their literacy skills?

## Janine Firmender

Co-PresenterSome of the supports I would recommend, based on some of our previous work and teaching experience, would be to scaffold the development of students’ mathematical writing. For example, when first engaging students in mathematical writing consider composing a class response with student input or having pairs of students write their ideas on sentence strips to share with the class. Additionally, engaging students in oral discourse related to the task or problem and helping students make connections between the oral and written discourse can be helpful.

## Madelyn Colonnese

PresenterBecause there has been limited work in mathematical writing, we identified participants for the task force who have experimented in this work across disciplines, in the classroom, or through curriculum and assessment development. We sought to bring together a diverse group of school- and university- based experts to ensure a comprehensive perspective and offer recommendations that would meet the needs of various stakeholder groups and students.

## Jeffrey Barrett

Hi,

I wonder about drawings and diagrams in relation to writing text about mathematics: how do more graphical or spatial images play into the way that students benefit from writing about mathematics? Are these visual and textual approaches mutually supportive ways of communicating and reflecting on mathematics? Also, does it matter whether children are explaining, justifying, or exploring ideas in writing?

## Tutita Casa

Co-PresenterSymbolic notation, visual representations (e.g., graphs), and other drawings used by elementary students were considered by the task force. We in part define “mathematical writing” as including written text as well as these other forms of communication, if they are necessary to a students’ particular piece. Individual students may utilize these visual components to support any type of writing.

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