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by Laney Sammons
Table of Contents
Writing is a process through which we record our observations, our thoughts, and our insights. It is a process through which we reorganize our ideas, develop conjectures, and gain insights. It is not content in itself, but a means of exploring and expressing content. So why would we not write in math class? (O'Connell and Croskey, 2008, p. 60)
If writing about math is so beneficial, why, then, is it not more common in math classes? Teaching students how to write about math is a challenge for many educators. Just as there has been little clarity about the kinds of mathematical writing students should do, minimal guidance has been provided about how to teach it. Even when teachers know what kinds of writing their students should be doing, they must then figure out how to teach them to do it.
The traditional instructional focus in math classes has been primarily on calculation and procedural skills, with little emphasis on developing students' conceptual understanding or their ability to communicate mathematically. This is the kind of math instruction most teachers experienced when they were in school, and many tend to teach in the same ways that they were taught. Because most were expected to do very little, if any, mathematical writing, they have scant experience upon which to draw to teach those skills. Yet, if teachers do not teach them, how are students to learn how to write mathematically?
Many elementary school teachers teach writing as well as math, but there has been very little carryover of literacy instructional strategies into math classes. Too often, there is an almost invisible barrier between subject areas that seems to segregate teaching methods to specific disciplines. Those teachers who focus only on math often have had very little training in how to teach writing and have few resources to draw upon to support this kind of instruction. Most math texts now include mathematical writing tasks of varying quality, but rarely offer teachers adequate guidance in how to develop their students' writing skills.
It makes sense for math teachers to look to literacy instructional strategies for ideas in teaching mathematical writing. Mathematical writing is, after all, a subset of writing in general. With recent changes in writing standards, there is an increased focus on both informative/explanatory and argument/opinion writing. Most mathematical writing neatly fits into one of those two categories. This means, of course, that in self-contained classrooms, mathematical writing may be taught not only during math class, but also during the language arts period, if desired.
The importance of supportive writing instruction is especially clear to kindergarten teachers. While some young learners arrive in kindergarten able to print the letters of the alphabet, write their names, or perhaps spell a few simple words correctly, even those few usually know very little about the actual process of writing to express meaning, as opposed to mere handwriting. With teacher support, these young students gradually discover that they are indeed capable of communicating their ideas with illustrations and even inventive spellings. Because few students of any age have had much experience writing about math, they face the same challenges in learning how to write about math as kindergarten students do in learning how to write in general. The same kinds of instructional support that teachers give kindergarten students are also effective in teaching learners to write about math.
Lucy Calkins (1994) spearheaded writing initiatives in schools with her work on writing workshops. She urged teachers to create environments conducive to writing in which students are respected and recognized as authors from a very young age. Implicit in her approach is the realization that students can write about their ideas even when they have not yet mastered the mechanics of writing. Her seminal work moved teachers to focus on written content, even when young authors can only communicate in writing with pictures or inventive spelling. While not minimizing the importance of writing mechanics, she encouraged teachers to celebrate the ability of students to tell stories or share information through writing, even when their work has spelling, grammar, or punctuation errors. In fact, because these young authors are excited about writing, they become more receptive to learning about these conventions of writing.
Math students benefit from the same inviting approach to writing. When their mathematical ideas are respected, even very young learners are eager to share them, whether orally or in writing. Teachers have the responsibility of not only welcoming early efforts at mathematical written communication, but also providing meaningful instruction to help learners develop more mature writing skills. Beginning in kindergarten, teachers should make writing an integral part of math class. Not only should they ask students to write about their mathematical ideas, but they should also provide guidance to show students how to do that more effectively.
A combination of three instructional modes—showing, sharing, supporting—has proven effective for teaching students to write (see Figure 5.1). Using these modes, teachers gradually release the responsibility for mathematical writing to their students (Pearson & Gallagher, 1983).
During the showing mode of instruction, teachers are primarily responsible for writing. They provide explicit instruction through minilessons or by modeling. Students begin to assume responsibility in the sharing mode as they work jointly with teachers to write about math and then revise writing to improve it. Finally, in the supporting mode, students write independently, either individually or in collaboration with other students, as teachers monitor their work, offer feedback, and conduct conferences with them.
The three modes of writing instruction can be ongoing, sometimes occurring simultaneously, depending on the types of writing students are doing, the writing strategies they are learning, and the mathematical content they are exploring. Obviously, students need considerable support when new writing strategies are introduced, but much less support when practicing previously learned strategies. Using a variety of modes of writing instruction also allows teachers to differentiate to meet individual students' needs (O'Connell, 2005).
Teachers can choose the level of support that best meets the immediate learning needs of their students. It is important to keep in mind, however, the strong rationale for gradually releasing responsibility to students during the learning process to help them become independent writers.
Most students only encounter mathematical writing in their textbooks. Textbook writing, though, is extremely specialized and limited. It is written for the sole purpose of making a mathematical concept or skill understandable to students. This kind of writing typically includes definitions and multiple representations of mathematical terms and concepts; it may also describe steps in a process or offer instructions for problem solving.
Although students do that kind of writing occasionally, textbook writing is hardly a suitable model for student writers who write to justify or explain their own mathematical ideas. Students need exposure to a rich variety of mathematical writing that encourages them to think more deeply about math as they write. In the showing mode of instruction, teachers offer explicit writing instruction that shows learners how to express their mathematical ideas.
In showing students, teachers may teach a minilesson about an aspect of mathematical writing, model the process of writing a short mathematical piece, or model the revision of writing done previously. This phase of instruction is completely teacher-directed; students are primarily listeners and observers. Because of this, the lessons are short and target only one or two teaching points. Writing strategies and tips are explained to students, who are then expected to practice them in their own writing. A minilesson on a writing strategy may be presented one day and then reinforced a day or two later with modeling of the same strategy. On subsequent days, the teacher might model the revision of the piece that was written.
Minilessons are highly focused and very brief. The teacher delivers them to either a small group or a whole class of students. Each lesson is designed to provide students with a writing skill or strategy that they can apply to enhance their mathematical writing. In no more than 10 minutes, teachers deliver a teaching point and give students a bit of practice in trying what was taught or a chance to talk about what was just demonstrated, so that they will be able to apply it independently in their own writing. Calkins (2005) suggests a specific architecture, or structure, for effective minilessons, which has been adapted to facilitate mathematical writing:
The following sample lesson scenarios offer an idea of how minilessons may be used to improve students' writing.
In this lesson, students learn how to use a math word wall as a resource when they are writing about their mathematical thinking.
Making a connection. Mathematicians, I am excited to see the writing that you have been doing. You have learned that you can share your math thinking using pictures and words. Today you are going to learn about something right here in the classroom that will make your writing easier for readers to understand.
Teaching. I am going to teach you how to use our math word wall to help you when you are trying to find the best math words to use in your writing or to spell the words correctly.
Today, I am going to write about how I plan to solve a problem. I am going to have a party and want to be sure that I have enough chairs for everyone. I know I have eight chairs, because I counted them. I have invited two families to the party. One family has three people, and the other has four people. Here is what I have written so far:
(The teacher displays a handwritten chart with the words:)
I know that I have a group of three people and a group of four people coming to my party. I have eight chairs. I have to find out how many people are coming all together so I will know if I have enough chairs.
(… and below the words includes a drawing of stick-figure people—one group of three and one group of four.)
I wrote about what I know and what I have to find out. Now I have to explain what I am going to do and why. Because I am putting groups together to find out how many, I plan to use addition to find the answer. I have two problems, though. I am not sure how to spell addition, and I can't remember the math word for the answer to an addition problem. Like all mathematicians, I want to be sure that readers understand my thinking, so I want the writing to be clear.
Let me think about this problem for a minute, and then maybe I can figure out what I can do. I remember that we have a math word wall! That will give me the information I need. Look at the word wall. There is the word "addition," so I can spell it correctly in my writing. I also see the word "sum." Now I remember—that's what the answer to an addition problem is called. I can use those words as I write about what I plan to do, and it will be clear to anyone who reads it.
Whenever you need to find out how to spell a math word correctly or to find a math word that you are having trouble remembering, you can look at the math word wall.
Actively engaging learners. Now, let's make believe that we are writing about a problem involving subtraction. Wow, that is a hard word to spell! We also need to know what the answer to a subtraction problem is called. Think for a few minutes without talking. How can you find out how to spell the word "subtraction" and find out what the answer to a subtraction problem is called? Now turn to a partner and share your answer.
The teacher listens to students as they talk to be sure they are using the math word wall as a resource and to identify students who still may not understand how to use the resource. After a few minutes, the students are called back together to discuss some of the ideas that were shared. Rather than calling on students at this point, the teacher shares the important ideas that were overheard, highlighting the parts of their conversations that are most important for them to remember. The students have already had an opportunity to be actively engaged. Now, the teacher can target the most significant points of their conversations, rather than having them muddled by student comments that may lack focus. It also keeps the minilesson brief, so students can spend more time writing independently after the minilesson.
As you talked, I saw so many of you point to our math word wall and then discuss how you can use it to find out how to spell math words and to find the correct math words to use in your writing. You figured out how to spell "subtraction" and what the answer to a subtraction problem is called.
Linking to future work. Now, mathematicians, remember that whenever you need help spelling a math word or finding the right math word, you have a valuable resource right here in the classroom. Just look to the math word wall! When we write in our math journals today, I am going to be looking to see who is using the math word wall.
The focus of this lesson is the importance of providing evidence to support ideas shared in mathematical writing.
Making a connection. In the last few days, we have been looking at some sample addition problems involving fractions with unlike denominators and writing about whether we agree with the sums that are shown. Most of the writing I have read clearly expresses whether or not you agreed. Today, I am going to teach you something very important that mathematicians do when they write about whether or not they agree with a solution or a mathematical idea. They offer evidence to support their thinking—specific reasons why they believe what they do that will convince others that their thinking is valid.
Teaching. Let's look at this piece of student writing.
The teacher displays the following sample to the class.
In this problem, they said that when you add ^{1}/_{4} and ^{2}/_{3} you get ^{3}/_{7}. That is wrong! They didn't add it right. The answer should be ^{11}/_{12}.
When I looked at this, I wondered how this student knew the answer was wrong. Nothing in the writing showed me how he knew that ^{3}/_{7} was incorrect.
When mathematicians write, they tell not only what they think, but also why they think the way they do and give evidence to support their ideas. So I thought to myself—what evidence could this writer give to justify this conclusion? Here is some evidence that I thought of to show that the solution was wrong.
First, I know that ^{2}/_{3} end sample is greater than ^{1}/_{2}. If I add ^{1}/_{4} to ^{2}/_{3}, it has to be even greater than ^{1}/_{2}. But ^{3}/_{7} is less than ^{1}/_{2}, so the sum cannot be ^{3}/_{7}.
Also, I know that, when you add fractions with unlike denominators, you have to find a common denominator. In this problem, it could be twelfths. The addition problem would then be ^{3}/_{12} plus ^{8}/_{12}, and the sum would be ^{11}/_{12}.
Do you see that I shared mathematical ideas to show why the sum is not ^{3}/_{7} ? I didn't just say it was wrong without telling why I thought so. Remember, whenever you share your mathematical thinking, it is important to give the reader evidence to support what you think.
Actively engaging learners. Let's practice. Here's another piece of student writing.
(The teacher shows this sample to the class:)
Four people plan to evenly share a pizza that has been cut into sixths. I have to find out how much pizza each person will get. I think that if it is divided evenly, each person will get three smaller pieces of pizza, or ^{1}/_{4} of the pizza.
The writer did not include any evidence to justify that answer. Talk with a partner for a few minutes about the problem and about what kinds of evidence you could give to justify the solution suggested in this writing.
The teacher listens to students as they talk to be sure that they are correctly interpreting the problem and are focused on considering ways to justify the ideas in the writing. After a few minutes, the teacher calls the group back together. Rather than calling on students to share what they discussed, the teacher restates some of the important ideas mentioned by students and draws representations to make them clear to all. Students have had an opportunity to talk about evidence that could be provided. Now, the teacher wants to be sure that all students hear some of the most helpful ideas. By sharing what was overheard, the teacher keeps the lesson focused and brief.
As you talked, I heard lots of ideas about how to approach this problem. Some of you talked about what you know about fractions—that if a whole is divided into four equal parts, each of those parts will be ^{1}/_{4}, so each person would get ^{1}/_{4} of the pizza. Some of you went further to say that the denominator in ^{1}/_{4} represents the number of equal parts into which the whole is divided, and the numerator represents the number of those parts. I also heard that there are four people, and each would get one of the parts. That also helps justify the response.
Others mentioned that the pizza-eaters would have to be creative as they divided up the pizza, since there were only six pieces. Each person could eat one of the pieces (^{1}/_{6}), and then two pieces would be left over. Those could each be cut in half, so each of these pieces would be ^{1}/_{12}. Everyone would get one of them. All together, each person would get ^{1}/_{6} and then ^{1}/_{12}, which added together is ^{3}/_{12}.
Another idea I heard mentioned was that if each of the sixths could be cut in half before anyone had pizza, then each person could have three of those pieces, which would be ^{3}/_{12}.
Linking to future work. As you and your partner discussed ways to justify the solution to this problem, you were working like mathematicians. Remember the kind of thinking you were doing, and whenever you write about your mathematical ideas, provide that kind of evidence to justify them for those who read your writing.
Students in this lesson learn how to brainstorm as a prewriting strategy.
Making connections.In the last few weeks, you learned that it is important to clearly identify ideas you are going to write about before you begin writing, so that your work is focused and not scattered. Today, I am going to teach you a prewriting strategy for thinking about what you already know about a topic before you begin writing.
Teaching. When I am writing, I find it helps me to have in mind many of the things I plan to say before I actually begin writing. Many mathematicians go through a prewriting process of brainstorming before they write. For example, if I am going to write about the Pythagorean theorem, I take time to think about what I know and what I might want to write about it. As I do, I jot those ideas down to help guide my writing.
(The teacher displays a piece of chart paper or a list on an interactive whiteboard on which the following terms and phrases are written:) hypotenuse, right angle, right triangle, square, triangle, equation, right triangle, formula to find the length of a side of a right triangle if the lengths of the other sides are known, and to determine if a triangle is a right triangle if the lengths of the three sides are known. (The teacher decided that having the words already written on a chart would be as effective as thinking aloud and recording them as part of the lesson, but would require less time.)
As I reflected on what I know about the Pythagorean theorem, these words and phrases came to mind, so I wrote them down. These words are giving me ideas about what I might write about the theorem. It takes only a few minutes, but the process of brainstorming gives me lots of ideas and some direction before I actually begin to write.
Actively engaging learners. I want to give you a little practice with this strategy. Working with a partner, brainstorm the words you think of having to do with the word equation. Jot down anything that comes to mind.
As students brainstorm, the teacher circulates around the classroom, listening to students' ideas. After a few minutes, the teacher calls the class together to discuss the brainstorming process. He shares some of the ideas that he overhears, rather than calling on students. This permits him to highlight the most important ideas and keep the lesson brief and focused.
It was wonderful to hear so much math talk going on! As I walked around the room, I heard you mention words like equality, relationships, formulas, equal signs, linear, quadratic, and much more. I even heard some of you talking about non-examples, like inequalities and expressions. It can really help your writing when you take time to brainstorm before you begin to write. The ideas you come up with help you decide what you are going to write.
Linking to future work. Whenever you are getting ready to write about a mathematical topic, remember to take time to brainstorm first. Jot down some notes to guide your writing, just as mathematicians do.
In using minilessons to improve students' mathematical writing effectively, teachers should keep in mind these recommendations (adapted from Collins 2004, pp. 27–29):
For all students, including those who are attempting to put their ideas down on paper for the first time, those who are refining their general writing skills, and those who are learning to express their mathematical ideas in writing, some of the most productive learning experiences result from observing teachers as they model the writing process and speak aloud about their thinking as they do. Students not only see what is being done, but also hear what writers think about as they write. Modeling and thinking aloud by teachers demystifies the writing process, making it accessible to learners.
Teacher modeling may focus on simple topics, like helping beginning writers understand the importance of leaving spaces between words, or on more complex compositional strategies, such as the use of word banks, the logical sequencing of ideas, the justification of mathematical thinking, or any other strategies that lead students to more clearly express their mathematical thinking in writing.
The modeling of writing by teachers as they share their thinking aloud is most effective when it is teacher-directed. Students should primarily be participatory listeners as they observe the modeling. It is often best to ask students to hold any questions they may have until after the modeling is complete. Although student questions deserve answers, if they are asked during the demonstration, they may distract from the instructional focus. In addition, when the questions are pertinent, they are frequently answered as students watch and listen. Any remaining questions may be addressed after the modeling and thinking aloud have been completed.
The writing process. To demonstrate to students strategies for writing about mathematical thinking (noting that the modeling process should last no longer than five or ten minutes):
The revision process. It is important that students have opportunities to see not only the initial writing process, but also the revision of previously written work. Although not all pieces of writing call for rigorous revisions, students need to appreciate the value of rereading to improve the quality of their written product. This type of modeling is too frequently omitted. As a result, students are reluctant to revise their work or struggle with the process when they do.
To model the revision of mathematical writing:
Effective modeling/thinking aloud. As you implement modeling/thinking aloud sessions:
In sharing the writing process, "teacher and students compose collaboratively, the teacher acting as expert and scribe for her apprentices as she demonstrates, guides, and negotiates the creation of meaningful text, focusing on the craft of writing, as well as the conventions" (Routman, 2005, p. 83). As they share the writing process with learners, teachers build upon what they have already modeled and provide a valuable scaffold for students who are learning to write about their mathematical thinking. With this mode of writing instruction, teachers not only share their own ideas, but also invite students to contribute to the writing process. Students assume some responsibility for writing, sharing it jointly with teachers. Because the teacher serves as a scribe, students are relieved of the responsibility for physically recording their ideas, so that they can focus on thinking about what to write and how to express it.
Teachers address specific learning needs of their students by exploring mathematical ideas and discussing possible writing strategies. With shared writing, teachers invite students to engage in writing experiences in which the entire writing process is made visible and concrete in a safe, supportive environment ("Shared Writing," n.d.). Students gain both competence and confidence on the way to becoming independent mathematical writers. Writing collaboratively with their teachers and their peers, students practice applying newly learned writing strategies.
In shared writing, shared revision, and shared interactive writing, a teacher "expands on the students' ideas, paraphrases their thinking, and demonstrates what cohesive writing looks like and sounds like" (Routman, 2005, p. 85). This is a valuable teaching strategy for all writing, but especially for teaching mathematical writing, when students have so little prior experience to draw upon.
Teachers and students share the task of creating a new piece of mathematical writing during shared writing. To effectively teach a shared writing lesson:
Shared revision is similar to shared writing, but instead of collaborating to compose writing, teachers and students work collaboratively to improve previously written pieces. Students and teachers:
Interactive writing is a form of shared writing in which teachers and students share the pen. It should only be used occasionally when there is a compelling reason to do so, because it considerably slows down the writing process. It may be used effectively to address a specific writing strategy with a small group of students who share the same learning needs or with very young students to help them gain confidence in their ability to write, including the physical act of recording their ideas.
Even when sharing the pen during interactive writing, however, teachers should be the primary scribes. Students should be asked to write only a single word or symbol at a time, rather than entire sentences.
Teachers should follow the suggested steps for shared writing and shared revision, except for calling upon students to record words, numbers, or other symbols as needed throughout the process.
To effectively share the writing process:
During this phase of instruction, students write independently after progressing through a learning process in which they gradually assume greater and greater responsibility for their mathematical writing. By this time, learners should have acquired the knowledge they need to apply newly learned strategies successfully as they write on their own. Of course, experienced teachers know better than to just turn their students loose at this point; students still require plenty of support to become proficient at writing about their mathematical thinking. Teachers should continue to closely monitor and support their independent writing.
Independent writing is probably the most common form of mathematical writing that students are asked to do in school. Unfortunately, students are often asked to do this without sufficient preparation for the task. Students are more successful when they first learn what the writing process entails and the characteristics of effective mathematical writing. Mathematical writing tasks and prompts that are assigned for independent writing should allow students to practice the writing strategies and mathematical content they have been or are being taught.
By closely monitoring students' writing progress, teachers can confirm that learners are prepared to write successfully before assigning independent writing. Being prepared, however, does not necessarily mean they will write successfully. Teachers can expect even well-prepared students to struggle as they write and practice new writing strategies; that is an important part of learning. However, being prepared means that students have a foundation of knowledge about writing and math that they can access as they write. To write well, students need ample practice with supportive coaching from their teachers.
Conferences are ideal occasions to learn more about students' thinking as they write and to prompt them to reflect on how to express their thinking most effectively. Supportive coaching offers ongoing encouragement, as well as timely and specific feedback to students as they write. Feedback is most effective when it occurs as students write, so they have the chance to implement it in the same assignment.
Teachers' observations as their students write about math are valuable formative assessments that can be used to identify and target additional learning needs. If many student writers are struggling with similar problems, teachers should make note of them and then give additional instruction to the whole class. For small groups of students who share the same needs, reteaching or additional instruction may be done in a small-group lesson. The needs of individual students can often be handled with one-on-one math writing conferences.
Collaborative mathematical writing is a form of independent writing in which two or more students work together to compose a piece of mathematical writing. The collaborative process encourages students to communicate their mathematical thinking and writing strategies orally with others, clarify their thinking, learn from one another, and work as a team. As teachers listen to their students talk, they gain valuable insight into students' thinking.
There are drawbacks to this kind of independent writing, however. If more than two students are working together, there is an increased probability that they will stray from the task. Less assertive or less engaged students may step back from the task altogether, leaving the writing process up to others. Furthermore, collaborative writing is generally more time-consuming. Therefore, while collaborative writing does have its benefits, it is advisable to assign students to write collaboratively only on rare occasions.
In a mathematical writing conference, teachers converse one-on-one with students about their mathematical writing (Sammons, 2014). These conversations are opportunities for students to share their thinking regarding both math and writing. Teachers sit next to students, conversing writer-to-writer. Teachers express their interest in their students' work, ask questions to clarify their understanding of it, and try to identify logical next steps in learning for their students. They provide immediate feedback that allows students to see their writing through the eyes of a more experienced writer. Teachers are also able to use conferences to provide brief and highly focused teaching points that students can begin applying immediately.
An effective conference lasts no more than about five minutes and is composed of four parts: conducting initial research to learn more about a student's mathematical thinking and writing process, making a decision about what is needed by the student, delivering a teaching point, and then making a link to the future writing by students (Calkins et al., 2005). Teachers should follow these principles when conducting mathematical writing conferences:
In this conference, a student is responding to this question regarding pattern blocks: If a trapezoid represents ^{3}/_{2}, which pattern block represents a whole? (derived from Thompson et al., 2008, p. 93)
Teacher: How is your writing going today, Matt?
Matt: Pretty good, I guess. I think I have answered the question, although it was kind of odd. I had to think backwards from ^{3}/_{2} to find out what a whole would be. It was hard to write about what I did.
Teacher: Tell me about your writing and the problems you had.
Matt: Well, I knew a whole was a parallelogram. I could just see it in my head, but I'm not sure what I wrote tells readers what I saw. This is what I wrote. See? "I know that I can put three triangles on the trapezoid. Since the trapezoid is ^{3}/_{2}, each triangle has to be ^{1}/_{2}. A whole must be a parallelogram."
The teacher's conference research indicated that Matt had a good conceptual understanding of fractional parts. He also used the correct mathematical terms for the pattern block shapes, but his written explanation was far from clear.
Teacher: You seem to understand the problem, Matt. In your writing, you used the correct math terms for the shapes of the pattern blocks. That makes it much easier for a reader to understand your writing, but I see what you mean about it not being as clear as it could be.
The teacher decides to teach Matt to include both words and diagrams in his writing to add clarity.
Teacher: One thing that mathematicians often do when they communicate their thinking is use diagrams and representations to show what they mean. Can you think of a way that you could add a diagram in your writing to illustrate your thinking?
Matt: I guess I could draw a trapezoid and show how the three triangles fit on it. Oh, and then I could label each triangle^{1}/_{2}.
Teacher: Yes, that would clearly show that each triangle is ^{1}/_{2}. Would you leave it at that?
Matt: No. Maybe I should draw a really dark line around the two triangles that are next to each other so somebody reading it can see the parallelogram. Let me show you.
Matt draws this diagram (see Figure 5.2).
Matt: Then I can write, "^{1}/_{2} + ^{1}/_{2} = 1." That shows that the two triangles together represent a whole. The two triangles form a parallelogram, so that represents a whole.
Teacher: I see now. Your diagram will show readers what you mean. I wonder if you could also describe what you did in words to make your writing more complete. Remember that when mathematicians write about math, they often include a diagram or other representation to make their writing clearer to readers. Whenever you write, keep that in mind.
To support the writing process:
To help students acquire a comprehensive understanding of what constitutes quality mathematical writing, teachers should teach them what good mathematical writers do as they write. Individual lessons should focus on skills that apply to writing in general, skills exclusive to mathematical writing, and skills related to the revision process. Shorter lessons with very specific, explicitly stated teaching points tend to be more effective than broader, less-defined lessons. Both during and after a lesson, students should be able to identify the teaching point.
If a teacher determines that students need to work on improving the openings of their writing, a minilesson might focus on how to write an effective opening by offering tips for composing solid openings. Students might be asked to compare sample pieces of writings to determine which opening sentences were most effective and why. The teacher might model the process of composing the opening for a piece of writing. This teaching point could then be reinforced with a shared writing lesson. As a follow-up, students might be asked to practice writing effective openings as they work independently in the supporting mode.
Figure 5.3 contains specific teaching point suggestions that may be used for instruction in any of the three modes of instruction previously described—showing, sharing, or supporting.
Writing Skills
Math Content
Revision Skills
Teachers should be aware, though, that even when students learn these important writing skills, they still might not understand fully what good mathematical writing is. With a focus on minutiae rather than the writing as a whole, individual "trees" may well hide the "forest." Teachers should help students develop an understanding that quality mathematical writing is more than just a list of characteristics or a description of math concepts and procedures.
Effective mathematical writers recognize that their writing has a purpose, is aimed at targeted readers, and fulfills a function—the sharing of one person's mathematical thinking with others. It is, in essence, a transmission of information and ideas—sometimes to inform, sometimes to persuade. Students need opportunities to examine examples of mathematical writing with an eye to function. They gain a more accurate conception of what constitutes quality mathematical writing when given opportunities to read, assess, and compare multiples examples of this kind of writing.
It is important that students write about their mathematical thinking every day, even if they just briefly reflect about the mathematical work they did that day. Many teachers find that the most convenient way to have students record their writing is in a math journal. These bound notebooks are easy for both students and teachers to manage. If student writing is done over several days, students know just where to locate their earlier work, rather than searching for individual sheets of paper. Additionally, by having most of the mathematical writing in a journal, students, teachers, and even parents can see how students' writing skills and mathematical knowledge have developed over time. While there may be occasions when writing is best recorded in other formats, journals are recommended for daily writing tasks. Mathematical writing can also be recorded and maintained by students on notebook paper or worksheets provided by teachers and then stored in pocket folders or three-ring binders.
With the availability of digital technology, students might try writing in digital journals or composing online blogs. The more accessible the writing is for revision and for later review, the better, though. Writing that is composed online and then forgotten is not nearly as valuable a learning tool for students.
Prewriting tasks enhance the writing efforts of all students and are especially beneficial to struggling mathematical writers. These tasks lead students to reflect on the topic and organize their thinking prior to writing. Although prewriting tasks are more commonly used in language arts instruction and less frequently in math classes, their use by student writers leads to better products. Described below are just a few of the many easy-to-use prewriting strategies that students can apply before they write.
Brainstorming. Students jot down everything they think of that has to do with the topic. Once they have recorded these ideas, they highlight the ones they think are most important and plan to include in their writing.
Sequencing. This strategy may be used once students finish brainstorming or any other time that they have a list of ideas or steps to include in their writing. They review the ideas they plan to include and then order them in a logical sequence before they begin to write.
Using a graphic organizer. There are several different graphic organizers that students can use to help with brainstorming and/or organizing their thinking prior to writing. Concept maps lead students to reflect on everything they know about a concept or procedure before writing, which often results in a more complete and thorough written product. Venn diagrams are extremely helpful to writers who are comparing and contrasting mathematical concepts. Frayer models are useful in helping students define mathematical terms as they describe the term in words, with a nonlinguistic representation, examples, and non-examples. These are just a few of many graphic organizers that teachers can share with their students to use during prewriting.
Analyzing a step-by-step procedure. With this prewriting strategy, students analyze a mathematical procedure before they write its description. Something of a writing task itself, it prompts students to think through problem solving or computational procedures logically and write a brief explanation of each step. This analysis can guide students' writing, helping them with both the sequence of the steps and descriptions of what they did during each step (see Figure 5.4).
In a basketball game, Kaliah's team scored 65 points. Kaliah scored 20% of the points. How many points did she score?
Step
What It Means
The team scored a total of 65 points.
Kaliah scored 20% of the points.
I have to find out how many points Kaliah scored.
I made sure I understood the information in the problem and what I need to find.
α = the number of points Kaliah scored
I labeled the variable in the problem.
20% = 0.20
I changed the percentage into a decimal to make it easier to work with.
0.20 x 65 = α
To find 20% of something, I have to multiply it by 0.20. The product will tell me how many points Kaliah scored.
0.10 x 65 = 6.5, so 0.20 = 13
It is easy to find 0.10 of a number. I used mental math. Then I just doubled it to find out what 0.20 of the number is. The product is 13.
0.20 x 65 = 13
13 = α
α = 13 points
So, α = 13 points. That is how many points Kaliah scored during the game.
Source: Thompson et al. (2008).
Topics for mathematical writing sometimes arise organically when interest is piqued as the students confront challenging mathematical concepts or conundrums. Teachers encourage curiosity about and interest in mathematics when they turn student questions into writing prompts that the students can use to explore topics of mathematical interest.
When students do not initiate topics, teachers can suggest prompts or questions that address the writing process or math content, promote self-reflection, or even engage students in creatively expressing their mathematical knowledge. Figure 5.5 presents mathematical prompts (most of which are open-ended so that students can address them in diverse ways) and questions that can be applied across grade levels.
Writing About Writing
Writing About Math
Writing About Reflection
Writing Creatively About Math
Source: Pearse & Walton (2011) and Thompson et al. (2008).
A big motivator for any kind of writing is the recognition of oneself as an author—someone who has something of value to share. Teachers increase the motivation of students to write about math when they provide opportunities for writers to share their mathematical writing with others. In addition to motivation, one of the best ways to improve students' writing skills is to "give students a sense of authorship" (Calkins, 1994, p. 267). When they know that they are perceived as authors and that others are reading their writing, they are more willing to invest the time and effort needed to write about math well.
To create the sense of authorship, teachers may ask students to read their writing aloud to the class in an author's chair, publish their work, or offer it to other students for peer review. With any of these options, students gain the satisfaction of being recognized as authors, sharing their own academic creations with others.
Author's chair. Prominent in most writing workshop classrooms is an author's chair. Seated in the author's chair, students read their writing aloud to either the whole class or a small group of students. Listeners are encouraged to ask questions about the writing and respectfully share their feedback. Writers may request guidance from their audience regarding specific aspects of their writing that they hope to improve. Inviting writers to share their work in this way lets them know that their writing is respected and of value—that it is worth sharing with other learners.
Student authors who read their work from the author's chair also benefit from having an authentic audience to give them feedback on their writing. Listeners learn from hearing how other students think about math, from exposure to the writing strategies they are using, and from discovering more about ways to express challenging mathematical content in writing. Listeners also learn that their feedback is valued. They listen to what is read in a more analytical way when they know that they will be asked to respond to it.
Publishing mathematical writing. Literacy teachers understand the worth of publishing the writing of their students. Publication is a more formal way of recognizing the work of student authors than the use of an author's chair and, at all levels of education, publication recognizes quality writing. While publication tends to be done more frequently in the elementary grades, it is important to keep in mind the important role publication plays in college and graduate schools.
While publication can be quite an involved process, it does not have to be. The options for publication range from creating a hardcover book to simply attaching a piece of writing to card stock, bound around the edge with colorful tape. Some teachers collect student writing to compile in a parent newsletter or have their students create small "foldable" books. Also, with the availability of digital resources, student writing may be published online. Writing may also be published in the form of brochures, posters, essays, or even cartoons related to mathematical content (Routman, 2005).
Publication, of course, involves more than creating a final copy of the writing; the purpose of publication is making writing accessible to readers. For students, as for any writers, it is important that others read their writing and offer feedback. Teachers may choose to include students' published mathematical writing in the classroom library. Students often enjoy sharing their writing with other classes or with their parents. School libraries or media centers may allow the display of published student mathematical writing for browsing by other students.
To make publication of student writing an effective part of instruction, teachers should focus on publishing several short and easily assembled works, rather than on one elaborate publication. With the frequent publication of shorter pieces of writing, students can apply newly learned writing strategies and math content in a timely manner and more quickly come to regard themselves as published authors.
Peer review. Another way students can share their mathematical writing is through peer review. When a student has completed a piece of writing, it is submitted to another student, who reads it and provides written feedback. The peer reader may be selected by the student or by the teacher. The peer feedback should focus on both the writing process and the mathematical content of the writing, noting specifically things an author has done well, including any questions the reviewer has, and offering constructive critical feedback. It is important that this feedback go directly to the teacher before being returned to the author, so that the teacher can ensure that comments are constructive and not hurtful. Prior to establishing peer review in a math class, it should be introduced to students with very clear guidelines as to what constitutes appropriate constructive feedback and what kinds of comments should be avoided.
As you begin to incorporate mathematical writing into your math instruction, reflect on your students' writing experiences and answer these questions:
The following are concerns frequently voiced by teachers, followed by suggestions to address them:
―Maintain high expectations for what your students should accomplish and share them constantly with your students.
―In minilessons, modeling, and shared writing, show them the kinds of writing products you expect and how they can accomplish that kind of writing.
―Teach students specific criteria for quality writing and provide exemplars to show them what constitutes effective mathematical writing.
―Get students excited about mathematical writing by sharing your enthusiasm for it. Let them see how intriguing mathematical ideas can be. Celebrate when you read good writing by students. Spotlight it by sharing it with the class.
―Provide timely feedback as students are working, so they can improve their writing as they work, rather than having to come back to it later for revision.
―Have students engage in assessing their own work using specific criteria for quality work.
―Vary the kinds of writing you assign. Not surprisingly, students get bored with repeatedly being asked to do the same type of writing.
―Students may not know how they found the solution or why they took the steps they did to find it. If that is the case, it is a problem of mathematical content knowledge, rather than a writing process problem, and should be handled as such. In these situations, the writing task provides accurate formative assessment data that identified specific mathematical learning needs. Students who lack the mathematical understanding needed can hardly be expected to write about their thinking and require support that focuses on the math content with which they are working.
―Students may be reluctant writers. Although some learners have a strong understanding of the problem-solving process, they may not want to make the effort to explain it in writing. If that is the case, it is a motivation problem. Solutions for this kind of problem were suggested earlier in this section.
―Students do not know how to express their mathematical thinking in writing. This situation indicates potential problems with both mathematical vocabulary knowledge and writing skills. When students' vocabulary knowledge is weak, it is difficult to clearly write about mathematical reasoning and procedures. In conferring with these students, teachers should check to see if insufficient mathematical vocabulary knowledge is contributing to their writing problems. If so, Chapter 6 discusses ways to build the mathematical vocabulary of learners. If vocabulary is not an issue, provide more modeling of mathematical writing strategies, opportunities for them to participate in shared writing experiences, and short writing tasks focusing on newly learned mathematical writing strategies immediately after they are taught to ensure understanding and application by students.
―At the beginning of a writing task, be sure that these students clearly understand the question or prompt. It is helpful to read it aloud with these students and clarify any challenging vocabulary. Consider illustrating the prompt with pictures.
―Provide math talk cards that have relevant math vocabulary with nonlinguistic representations for each word and sentence starters that can be used when writing about the prompt. Review these cards with students before they write.
―If necessary, have students address the prompt or question by drawing pictures. Later, as you confer with them, have the students talk about what they drew. Serve as a scribe and record their words to accompany their illustrations.
―Offer authentic and specific praise for the efforts of these learners—not just "Good thinking!" or "Good writing!" Let them know that you recognize the efforts they are making.
―When you are planning math lessons, carefully select the most important practice problems and have students complete those. In lieu of assigning all the problems, have them complete writing tasks that relate to the content of the lesson.
―Review teaching materials to be sure that each lesson aligns with the grade-level standards you are teaching. Because most math teaching resources are sold in multiple states where math standards vary, some of the lessons included may not address standards for your grade level in your state. To maximize the time you have to teach the essential standards for your state, eliminate lessons that do not align with your standards.
―Mathematical writing tasks are ideal workstation tasks for classrooms where math workshops are being used.
―Incorporate mathematical writing tasks as formative assessments to determine the learning needs of your students. With traditional assessments, students may accidentally come up with the correct solution to a problem when they do not really understand the math. Conversely, an incorrect solution without an explanation about the problem-solving process by a student does not offer the assessment information needed to allow teachers to identify crucial learning needs. The student may understand the math, but made a simple error in calculation. On the other hand, the student may lack prerequisite background knowledge or may not have understood the problem.
To support students effectively, teachers need to know more than just whether a student's answer was correct. When students write about their thinking, it makes it visible to their teachers. Using that information, teachers are much more likely to be able to address their students' needs. Yet, unless students have ongoing opportunities to write about their mathematical ideas and have received instruction on how to do it effectively, assessment items requiring writing will be of little use.
As described in this and in earlier chapters, oral and written mathematical communication by students should be an integral part of mathematics instruction throughout all grade levels. Effective communication, however, is dependent upon a solid foundation of mathematical vocabulary knowledge, both receptive and expressive. Chapter 6 examines ways to help learners develop such a foundation.
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