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Give students a problem and a correct solution and let them know it is correct. Their job is to "prove it" by coming up with a convincing mathematical argument. Jo Boaler has three levels of convincing- yourself, a friend and a skeptic. Klapinsky expands it to more levels.

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6 examples to try in your classroom

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The 2002 and 2007 reports of the National Assessment of Educational Progress (NAEP) writing as ses sment (National Center for Education Statistics, 2008), administered to 8th and 12th graders, show an increase in writing scores. Applebee and Langer's (2006) analysis of NAEP data as well as other sources published during the decade preceding their review revealed that the more frequently students reported writing one or more paragraphs in science and social studies, the higher their writing achievement.

The results, however, were lower in mathematics. "It may be that at the classroom level, the role of writing in mathematics instruction has not been well conceptualized" (Applebee & Langer, 2006, p. 14). Indeed, many teachers find it more natural to integrate writing and science (e.g., Varelas, Pappas, Kokkino, & Ortiz, 2008) or writing and social studies (e.g., Jones & Thomas, 2006). Wolsey (2010) examined the complexity of student writing and vocabulary learning in a cross-disciplinary writing project involving English, science, and social studies. Where was mathematics?

That we see so few examples of the integration of writing and mathematics in educational literature seems surprising, considering that the mathematics education community has affirmed the importance of such integration for many years. As early as 1989, the National Council of Teachers of Mathematics (NCTM) identified learning to communicate mathematically as a major goal for students.

In 1992, Countryman's book, Writing to Learn Mathematics: Strategies That Work, K-12, captured the attention of mathematics educators amid a flurry of interest and ideas (e.g., Quinn & Wilson, 1997; Sipka, 1990). Almost a decade later, NCTM (2000) specifically stressed writing as "an essential part of mathematics and mathematics education" (p. 60) in the landmark document Principles and Standards for School Mathematics.

Still, many teachers struggle to link writing and mathematics and honor the integrity of both disciplines at the same time. Teachers of writing might say that if students are assigned to describe the process they used in solving a problem with no revision or editing, the quality of integration is drawn into question. Teachers of mathematics might say that if students are asked to write a report on a famous mathematician they may not be engaged in developing mathematical reasoning no matter how many drafts they write. Although the appropriate balance may be elusive, the endeavor is nevertheless worthy of being undertaken (e.g., Applebee, 1984; Bossé & Faulconer, 2008; Newell, 2008).

There are two levels of integration that teachers may use as a beginning point. Writing without revision, the first level, can be readily worked into mathematics instruction. Writing with revision, the second level, may take more time but enables teachers to connect the writing process more fully with mathematics instruction. Each level can be appropriate under differing circumstances.

In Carter's (2009) description of writing tasks in her mathematics instruction, both levels are reflected. She had her students write in what she called "mathematical notebooks" (p. 607), which required little or no revision. She also had her students write about mathematics in a process-oriented way during writing workshop. Some wrote stories such as "The Hexagon Adventure" and "The Magical Subtracting Frying Pan." Others wrote autobiographies such as "How I Learned to Do Math in Kindergarten and Beyond."

Although Carter's (2009) reflective mathematical journals and stories about mathematical concepts in writing workshop may not represent an ideal solution, they demonstrate one teacher's exploration of the possibilities. This article presents six additional examples, including student work, in which teachers have attended to the goals of both writing and mathematics. By no means are these ideas new. Teachers may have been introduced to them before, but perhaps not in the context of mathematics. Although the following suggestions are not exhaustive, teachers have found them to be a promising place to begin.

Writing without revision

Learning logs

As students start class, they are given a prompt to which they respond for a few minutes in writing. The task is not designed as a mathematics problem per se, but rather to encourage students to focus on mathematics.

One fifth-grade teacher typically used learning logs to review previously learned material. On one occasion, the prompt she posed was, What did we learn about mean, median, and mode? Most students wrote at least a half page filled with definitions and examples (see Figure 1A). While studying probability, one student wrote the following definition: "The 'probability' of something is how likly your gonna get, pick, or find something." He then gave an example using shapes and colored cards (see Figure 1B).

Figure 1A: Learning log entries by fifth graders - definitions and examples of mean, median, and mode

Figure 1B: Learning log entries by fifth graders - definition and examples of probability

This teacher found that the quality of the learning logs improved as students shared their work: "[When a student shared] a log that showed clear understanding, this provided others with an example to follow." Along with finding learning logs to be an effective way to introduce or close a lesson, the teacher observed that the quality of student discussion of mathematics during lessons was richer when students were expected to write.

Learning logs honor the integrity of writing when students write their own connections and examples. At the same time, they foster mathematical understanding by engaging the minds of students in transforming information from facts to be memorized to the construction of meaning (e.g., Newell, 2008).

Think-write-share

Teachers often ask questions and count on having at least one or two students raise their hands. One 2011fourth-grade teacher tried to involve everyone in the class by allowing think time and expecting students to write before sharing. For one think-write-share, she asked, "What is an equivalent fraction?" She gave the students a few minutes to think and then asked them to write their responses before calling on students to share, using a document camera. In reflecting on the use of this strategy, the teacher commented,

I assumed that everyone understood [what an equivalent fraction is], because we've done it for a while. However, as we shared, I soon realized that some of my students thought that equivalent fractions were [only] fractions that were the same, for example, 1&#8260;4 = 1&#8260;4.

To address this misconception, the teacher strategically selected students to share based on the thinking she had seen demonstrated: "Kids who were getting it wrong, kids who were getting it a little, and those who knew it well." Once several clear responses had been shared and discussed (see Figure 2A), the teacher invited all students to write again, revising their original work to reflect their increased understanding (see Figure 2B).

Figure 2A: Think-write-shares by fourth graders - example showing a clear understanding of equivalent fractions

Figure 2B: Think-write-shares by fourth graders - example of equivalent fractions, before and after revision

The think-write-share strategy heightens student engagement in writing. Concurrently, students are held accountable for their own mathematical understanding.

Note-taking/note-making

Students may be accustomed to taking notes, but now ask them to make notes as well. Along with listing the main points of a lesson, students can write their own reflections and perceptions. One fifth-grade teacher used the strategy of note-taking/note-making in mathematics to extend the notion of whole numbers to integers, including the concept of negative number.

She asked her students to fold their papers in half vertically. On the left side, she directed students to define integers and construct number lines to demonstrate relative size of integer pairs. On the right side, students wrote their own reactions and observations. One student wrote, "I will remember that on the negitive side the bigger it gets the smaller it is. I thought that was cool." Another student wrote, "It is weird that -2 is greater than -5." This same student went on to create his own analogy: "It is like a piano, lower to higher with middle C. Smaller to larger with zero. Zero and C aren't small, big, high, or low" (see Figure 3).

Figure 3: Note-taking/Note-making: A fifth grader's conception of integers

Note-taking/note-making honors both writing and mathematics. It encourages students to make connections between new concepts and previously learned material and their personal experiences.

Writing with revision

Shared writing

A third-grade teacher used shared writing with her students to review what they had learned in their geometry unit. Using chart paper, the teacher recorded a student-generated word bank that included face, edge, vertex, congruent figures, and polygons. On subsequent sheets, the teacher used a black pen to record in sentences what the students shared about their learning. Next she guided them to suggest revisions and additions to their shared writing and recorded those suggestions with a different colored pen.

Some changes related to word choice ("learning about geometry" became "learning and exploring geometry"), while other changes related to geometry concepts. In the shared writing, students had listed 3-D shapes such as spheres, cones, cubes, and cylinders and had given real-life examples. During revision, students added the following sentence that went beyond visualization to include attributes: "Some 3-D figures will have vertexes (corners), edges, and faces." Students then suggested three titles and voted on which title was best (see Figure 4).

Figure 4: A shared writing about geometry composed and revised by third graders and their teacher

This teacher valued shared writing as an engaging strategy to help students internalize mathematics:

I feel that this experience has helped them go beyond recording the information in their math journals and just doing our assignments.... I think it helped to make it more personal and [validate] what was important to them.

As students engaged in revising, they improved the writing while extending their mathematical thinking (see Whitin & Whitin, 2000). Because the mathematical ideas are constructed through group interaction, students help one another learn to communicate mathematically.

Class book

Using a similar process, a teacher can make a class book with students. Once a shared writing is completed on chart paper, the teacher numbers the sentences and assigns each student a numbered part to write as a final draft for a page in the class book. Very little time is required for the teacher to move from student to student and edit sentences. As the teacher edits, students are encouraged to create representations for the idea on their page.

When the students are finished, the teacher gathers the pages in order and adds a cover with a title. When the shared writing has only a few sentences, the teacher assigns several students the same number and creates several covers. In this way, students produce multiple copies of the same book. One could be kept in the classroom and others given to teachers of younger grades for their reading centers.

A fourth-grade teacher used a variation of this strategy with her students. This teacher's prompt was to create and illustrate a word problem using fractional parts of a set of 24 (see Figure 5). Every student's work became a page in the class book.

Figure 5: A class book page about word problems written and illustrated by fourth graders

Class books provide a sense of audience as well as an opportunity for students to revise and edit their writing. Simultaneously, the format invites students to "create and use representations to organize, record, and communicate mathematical ideas" (NCTM, 2000, p. 67).

Alphabet books

One fifth-grade teacher helped her students create alphabet books to develop mathematics vocabulary. She assigned students letters of the alphabet and sent them on a search for new and complex words in their mathematics textbook, notes, thesaurus, and mathematics dictionary (e.g., Monroe, 2006). Students were assigned to use each word in a meaningful sentence, draw an appropriate representation, or write a real-world connection (see Figure 6). The teacher explained,

Because the format of an alphabet book requires brevity, I can readily respond to what students have written individually in short conferences, offering both compliments and suggestions for improvement. I can discuss their mathematical thinking and their writing in the same conference.

Figure 6: A class book page about word problems written and illustrated by fourth graders

Students then revise and edit as they prepare a final draft for publication. This teacher prefers to have the book professionally printed, using an inexpensive online publishing tool (e.g., mightyauthors.com).

Alphabet books not only introduce students to a delightful genre, but also allow students to experience every phase of the writing process. At the same time, they honor the integrity of mathematics by helping students develop vocabulary essential for mathematical thinking. The format invites the use of meaningful communication through words, graphic representations, and symbols.

A call for additional strategies

These six ideas represent only a place to begin. Creative teachers will use these ideas as springboards to additional strategies for integrating their writing and mathematics instruction. Numerous and varied opportunities for this integration support students as they learn to think their way into mathematics and make it their own (Zinsser, 1988) and also become more aware of the range of writing possibilities available to them (Newell, 2008). Such opportunities are not superfluous or simply nice to do if there is enough time. They are too important to ignore.

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The 2002 and 2007 reports of the National Assessment of Educational Progress (NAEP) writing as ses sment (National Center for Education Statistics, 2008), administered to 8th and 12th graders, show an increase in writing scores.

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Going past "Explain your answer" to help students be better communicators

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One of reasons I became a teacher is that I get to share what I love about math with others. The beauty and the connections inherent in mathematics - these are things I am passionate about and I share that passion daily. When I see students who are engaged in my classes with that same level of passion, there is nothing that makes me prouder. Standard for Mathematical Practice (SMP) 3 - "Construct viable arguments and critique the reasoning of others" - provides an opportunity for us to see students exhibit that passion.

Many teachers believe that, by adding "explain your answer" to the end of a problem, they have covered SMP 3. While the addition of this phrase is a step toward constructing a viable argument, it is not attending to the full meaning of the standard. Our kids can do more and should be given opportunities to do more.

We can't (or at least shouldn't) create arguments out of thin air. Creating a thoughtful, evidence-based, unassailable position statement is a skill that we continue to hone throughout our lives - a skill that is critical to any career. In math, students need to learn how to create an argument: what constitutes evidence, how to ask questions to get more information, and how to analyze multiple examples to inform their argument. These are all components of SMP 3 and should be practiced even in the earliest grades.

In elementary grades, opportunities to practice SMP 3 could be based on the operations that students are working with. Below we see an opportunity for students to create an argument as well as a chance for students to critique the reasoning of others. These are different, yet very much interconnected, skills and instructional materials should prompt students to do both.
First question was adapted by the author. Second question is by Illustrative Mathematics: https://www.illustrativemathematics.org/content-standards/5/NBT/A/2/tasks/1524
Looking at the examples, we begin to see more clearly the numerous skills that students must develop to be able to truly practice SMP 3. These include:

Using prior results to develop arguments

Providing counterexamples

Justifying conclusions

Identifying and explaining flaws in arguments

Using questions to clarify reasoning

Comparing multiple arguments to determine which is more effective

Created by the author From Illustrative Mathematics: https://www.illustrativemathematics.org/content-standards/7/RP/A/tasks/1589
From Engage NY Grade 4: https://www.engageny.org/resource/grade-4-mathematics
We can see how we are extending these skills as we look at work that students are doing in later grades.

As we have seen, instructional materials can provide opportunities for students to construct arguments. A question can be presented as a request for students to justify their reasoning or as an argument for students to critique. Students can also be presented with multiple arguments and have to determine which is correct or more effective. While all of these problems do address key skills required for performing SMP3, the bigger theme of the math practice is being an effective communicator. Just because materials ask for an argument and one is written, it doesn't mean that the argument is logical and makes sense to others. It is possible to learn how to structure arguments by critiquing, but to meet the full meaning of the math practice you need the feedback that comes from interacting with others and having them respond to the argument you present. Additionally, that interaction is the only way to find opportunities to ask clarifying questions

We want instructional materials to provide scaffolds and potential opportunities to attend to the full meaning of SMP 3, with the understanding that the structure of the course must be conducive to students engaging in discourse. This discourse will provide time for students to develop the skills of creating logical arguments and an understanding of how to ask quality clarifying questions.

It is important to remember that a handful of these opportunities sprinkled throughout an entire year will not be enough to get students to share their excitement and passion for the math task they are engaged in. In my experience, the more opportunities students have to engage in this work, the more confident they become in sharing their understandings and reasoning. It is then that students will show a deeper passion for the work they are doing.

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Digging Deeper into SMP 3 - Construct Viable Arguments and Critique the Reasoning of Others - Achieve the Core Aligned Materials

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One of reasons I became a teacher is that I get to share what I love about math with others. The beauty and the connections inherent in mathematics - these are things I am passionate about and I share that passion daily. When I see students who are engaged in my classes with that same ...

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Standard 3- Connections to Classroom Practices

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Standard 3- Connections to Classroom Practices

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Grade specific videos of lessons relating to students constructing viable arguments and critiquing the reasoning of others.

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classroom observations:
Teachers who are developing students' capacity to "construct viable arguments and critique the reasoning of others" require their students to engage in active mathematical discourse. This might involve having students explain and discuss their thinking processes aloud, or signaling agreement/disagreement with a hand signal. A middle childhood teacher might post multiple approaches to a problem and ask students to identify plausible rationales for each approach as well as any mistakes made by the mathematician. An early adolescence teacher might post a chart showing a cost-analysis comparison of multiple DVD rental plans and ask his students to formulate and defend a way of showing when each plan becomes most economical. A teacher of adolescents and young adults might actively engage her students in extended conjecture about conditions for proof in the construction of quadrilaterals, testing their assumptions and questioning their approaches. Visit the video excerpts below to view multiple examples of teachers engaging students in formulating, critiquing and defending arguments of mathematical reasoning.

the standard:
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and-if there is a flaw in an argument-explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

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common core resources /mathematical practice standards /standard 3: construct viable arguments & critique the reasoning of others /

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classroom observations: Teachers who are developing students' capacity to "construct viable arguments and critique the reasoning of others" require their students to engage in active mathematical discourse. This might involve having students explain and discuss their thinking processes aloud, or signaling agreement/disagreement with a hand signal.

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Developing Reasoning

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with special thanks to Christopher Danielson and his Which One Doesn’t Belong - A Shapes Book.

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with special thanks to Christopher Danielson and his Which One Doesn’t Belong - A Shapes Book.

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Helping students learn how to convince others that something is true

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The Standards for Mathematical Practice (SMP) in the Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) highlight the mathematical habits that educators should be fostering in mathematics classrooms throughout K-grade 12 education.

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This teaching strategy may help support students.

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Ruth covers some important topics when we think about writing in general. There are several copies of the book in our building.

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children with social disabilities - education

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composition (language arts) - study and teaching - psychological aspects

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english language - composition and exercises - study and teaching - psychological aspects

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biography & autobiography / personal memoirs

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In this moving, personal book, Ruth Ayres weaves together her experience as a mother, teacher, and writer. She explores the power of stories to heal children from troubled backgrounds and offers up strategies for helping students discover and write about their own stories of strength and survival...

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A new era of standardized testing is quickly approaching with the dawn of the Common Core Standards assessments in the fall of 2014. And districts all over the country are adjusting their curriculums to meet the challenge. The Common Core requires students to think and learn in a much deeper way, and one of the best ways to facilitate that deeper learning is to get kids writing. Not just in English class, but all the time.

"Writing is an absolute imperative for success with the Common Core," said Steve Peha, founder of Teaching That Makes Sense, a national education consultancy specializing in literacy, leadership and school-wide change. "The Common Core and its associated tests set a significantly higher bar for student achievement. When you include the Next Generation Science Standards, it's pretty easy to see that kids are going to have to be much better writers than they have ever been before. Writing regularly, in all subject areas but especially in math, social studies, and science is going to be crucial."

What Is Writing Across the Curriculum?
Writing Across the Curriculum is a movement that began in the 1970s and is gaining a lot of attention these days. It is designed to boost children's critical thinking skills by requiring them to write in all of their
classes-from math to social studies to science-and not just in language arts. And it's a perfect fit for Common Core.

The new standards will require that content area teachers reinforce the benchmarks that ELA teachers traditionally have covered in their classrooms. This means that the burden of literacy will shift to the entire teaching staff. Math, science and history teachers will be tasked with using writing assignments to help kids learn the subject matter at hand, whether it's photosynthesis or algebra. Going forward it will be more important than ever that teachers coordinate their lesson plans in support of the Common Core Standards.

Why Write Across the Curriculum?
Learning to write, and write well, is a crucial life skill. Not only does it help one succeed in school, it's vital to success in the "real" world too. Writing has become essential in today's wired world. We communicate through the written word on a daily basis via email and text.

And when today's elementary and middle school children enter the work force, they will be expected to know how to fire up their computers and write coherent, intelligent content pretty much no matter what field they choose.

In addition, studies have shown that writing helps boost student achievement across the board because it actively engages children. It requires them to take in information, organize their thoughts, sort through all of the information they've received and then process it. "Writing is the most extensive brain workout a kid can get," said Mr. Peha. "It includes reading, logic, motor coordination and if writing for an audience, social emotional intelligence." In other words, there are a lot of benefits to writing:

Writing aids retention. It helps children remember and understand material much more than passive forms of learning like reading and listening. "Writing helps with mental recall," said Mr. Peha. "It cements a concept in a kid's brain more than anything else."

Writing increases the depth of knowledge on a subject and helps students master any specialized vocabulary or terminology related to the topic.

Writing develops critical thinking skills. When a child has to research something, organize her thoughts and then write about it in a clear, concise way she's flexing her mind.

Writing promotes independent thinking. In order to write, you have to have a point of view.

Writing Across the Curriculum Benefits Teachers
As daunting as writing across the curriculum may sound to some teachers, there are a lot of positive things about incorporating writing into your lesson plans!

Writing is a great way to engage allof your students! You won't just hear from the kids who are always throwing their hands up.

Writing helps teachers monitor student progress and gauge their strengths and weaknesses. "Writing lets you know where your students are at more than any test," said Mr. Peha. With regular writing assignments as part of your curriculum, you don't have to wait for a big exam to see if the kids are getting it.

Writing helps you see gaps in instruction so you can adjust your teaching to insure that all of the students get what you're talking about!

Writing saves you time! Writing can be a very efficient way to cover multiple standards at once because it is such a complex, multifaceted task. In science, for example, you can get a glimpse of students' hypothetical and procedural thinking by analyzing a written lab report.

Students learn best by writing. "Kids learn the most through writing," said Mr. Peha. "If you're only going to do one thing, writing is the highest payoff activity you can pick."

Getting Started With Writing Across the Curriculum
Some teachers have expressed concern over writing across the curriculum. Math and science teachers may worry they'll have to spend an inordinate amount of time teaching writing skills and have to become experts in the six traits of writing or give up their evenings to grading stacks and stacks of papers. But that's not the goal of writing across the curriculum. The point is deeper learning, not a perfectly developed writing product as one would aim for in English class. There are many ways to incorporate writing into lesson plans without requiring a teacher to become a six traits whiz.

Here are seven awesome ways to bring writing into your classroom no matter what subject you're teaching:

1. Journal Writing: Journal writing is a great way to create confident writers. Journals are an informal place for students to summarize their thoughts and think about class content, no matter what the subject. You can give the children writing prompts or just let them write freely!

2. Think-Pair-Share: After a lecture or presentation, invite the children to record their thoughts. Then pair them up with another student and have them discuss the topic. Finally, open the discussion up to the whole class. You'll find that by organizing and writing their thoughts before the discussion, the kids will have much more insightful things to add to the conversation!

3. Quick-Writes: Quick-writes are great ways to get students to practice writing and critical thinking skills. They're designed to focus the student's thinking. Set a timer for 10 minutes and give the children a writing prompt. You can show them an historical picture, read a quote from your favorite scientist or ask them to write about how they'd use a math theory in real life. Anything that gets them thinking...and writing! Not only are quick-writes quick to write, they're incredibly quick to review as well. Short writing is going to be as important as long writing with the Common Core Standards. All children will have to express coherent thoughts in both short and long time periods.

4. Self-Assessments: Throughout the year invite the students to write about how they think they're doing in class! Ask them what the most difficult part of the class has been or what they've loved learning. Not only will they get practice writing, you'll get valuable insight into how your students are learning and what you can do to help them even more!

5. Real World Writing: Think about the type of writing most often done in your discipline and have the students do it! For example, mathematicians write theorems and textbook problems. Scientists write lab reports. Journalists in all fields write articles. Have the kids create a website or a pamphlet for some real world writing experience. This not only gives the students hands-on experience in the discipline, but fulfills the Common Core requirement that students produce not only short writing assignments, but longer, more involved assignments too.

6. Note Taking: There isn't a ton of writing development in scribbling notes as a teacher is talking. But you can use note taking to flex your students' writing skills if you tweak things a bit. Mr. Peha loves using "summary note taking" as a writing exercise. He suggests breaking your lectures down into 5 to 10-minute chunks and inviting the students to summarize what you spoke about at the end of each block. They'll get to flex both their writing and retention skills and you'll get a break to catch your breath!

7. Research Projects: The Common Core Standards require all students to be able to research a topic in any discipline and write about it. So ask your students to write research-based arguments, not just persuasive arguments. The goal is for all children to become self-directed learners that are adept at researching (and writing about!) a wide variety of subjects.

Writing across the curriculum is a wonderful way to get kids writing and learning in bigger, better, deeper ways. Is your school working on writing across the curriculum? Please share what you're doing in the comments.

Looking For More Writing Help?Check out our collection of free writing across the curriculum resources!
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Writing Across the Curriculum: What, How and Why - WeAreTeachers

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By Deva Dalporto A new era of standardized testing is quickly approaching with the dawn of the Common Core Standards assessments in the fall of 2014. And districts all over the country are adjusting their curriculums to meet the challenge. [...]

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Using graphic organizers to create different kinds of writing

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Using graphic organizers to create different kinds of writing

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Mary Lee Hahn is a friend who teaches fifth grade in Dublin. I have a huge amount of respect for her work. This may give you ideas of how kids could write...giving them some choice.

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Being specific with feedback and narrowing your focus might help improve writing.

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Making the Most of Your Writing Feedback

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Giving high-quality feedback on student writing can be a challenge, but these strategies help maximize its impact on your students.