**Using Writing In Mathematics**

This strand provides a developmental model for incorporating writing into a math class. The strand includes specific suggestions for managing journals, developing prompts for writing, and providing students with feedback on their writing. In addition, the site includes two sample lessons for introducing students to important ideas related to writing about their mathematical thinking.

__Materials__paper and pencil

overhead pens

overhead projector

calculators (optional)

__Teaching Strategies For Incorporating Writing Into Math Class: Moving From Open-Ended Questions To Math Concepts__**Starting Out Gently with Affective, Open-Ended Prompts**

Writing about thinking is challenging. For this reason, it's best not to start out having students write about unfamiliar mathematical ideas. First get them used to writing in a math class:

- Begin with affective, open-ended questions about students' feelings.

__Sample Direction #1__: Reflect on your participation in class today and complete the following statements:

I learned that I...

I was surprised that I...

I noticed that I...

I discovered that I...

I was pleased that I...

__Sample Direction #2__: Describe how you feel about solving _________ problem.

- Have students write a "mathography"-a paragraph or so in which they describe their feelings about and experiences in math, both in and out of school. (This is a good tool to get to know students early in the year, and to make comparisons later when looking for signs of progress.) (see
)__Mathography Prompts__

- Find ways to keep students writing for the allotted time:

**Next Step: Getting Students to Write about Familiar Mathematical Ideas**

1. Once your students have become accustomed to writing about their attitudes and feelings toward mathematics in their journals, they are ready to write about simple, familiar math concepts. It is important not to make the writing too difficult by asking them to write about unfamiliar math ideas. Using writing to review familiar math ideas will increase confidence and skill in writing as well as revisit important math concepts.

Sample Directions:

Explain in your own words what subtraction means.Explain what is most important to understand about fractions. 2. Use student writing samples to help them refine their writing. (Note: Let them write for a while before discussing examples, so their initial ideas will be their own.)

3. Introduce the term metacognition to help students understand the reason and audience for their writing.

**Moving On: Writing About More Advanced Math Concepts**

When you feel your students are ready, ask them to write about more complex mathematical ideas, including concepts being taught at their current grade level. To help you move your students into this more advanced level of writing about their thinking. Here are some other suggestions to help you:

1. Encourage your students to use drawings and graphs to explain their thinking.

- Research shows that using simple visual aids (diagrams, graphs, etc.) improves mathematical problem-solving ability, especially in female students.

- Ask the group to write a summary of how they reached a solution, including any "false starts" or "dead ends."

- Ask each individual to write an explanation of the group's work on a problem. Have the small groups discuss the individual explanations.

- After a small group assignment, have students "explain and illustrate two different approaches to solving a problem."

__Getting Started with Two Lesson Plans__**Lesson Plan #1: "Metacognition"- Teaching Students to Think About Their Thinking**

**Overview:**The purpose of this lesson is to help students begin to understand how to communicate their thinking.

**Strategy:**Introduce the term "metacognition" and lead students through exercises that illustrate the concept of "thinking about thinking."

**Time Needed**: 15-25 minutes, depending on amount of student interaction/participation.

**Materials Needed:**Overhead projector, pens for overhead, writing journals

**Procedure:**

1. Introduce "metacognition" by writing it on the overhead and explaining what it means. Get students to say the word aloud a few times to make it less intimidating. Then you might tell a story about research in how the human brain works.

2. Tell students that the more attention we pay to our thinking, the more we'll come to understand about the process of thinking. Although we're used to just being concerned about the results or the "answers," if we pay more attention to how we think, it would help us to think more clearly, and improve the quality of our results.

3. As a non-threatening exercise to illustrate metacognition, ask students how they decided what to wear that day. Ask three or four students to share their answers out loud. Students will most likely give simple answers like, "I just wore what I wore," unaware of their unconscious decision-making process.

4. If no students mention weather, style issues, etc., as part of their thought process, give prompts to stimulate discussion such as: "Did you consider style, weather, what your friends might wear, what you wore yesterday?" Ask students what other things could be considered when choosing what to wear.

5. List their ideas on the overhead. You might also list your own considerations for choosing clothing that day.

__Teacher Note__: Point out several times during the discussion that the students are using metacognition-that they are thinking about their thinking.

6. Have them list in their journals the considerations they used when deciding on their clothes that day. Ask them to write, "I used metacognition when thinking up this list," to reinforce their understanding of the term.

7. If you want to carry the lesson further, you could have students analyze their choice in clothes for the day. Had they made the best choice? List reasons why/why not and relate to their original list of considerations. Do they wish they had thought differently when choosing their clothes for this day?

__End Result of Lesson__: Students will have a beginning concept to use in their discussions/writings about their math answers; and you, as a teacher, have the clothing example to return to many times as an example of metacognition they can then apply to math.

**Personal Narrative:**

**One Teacher's Experience with a Metacognition Lesson**

After reading many math journal entries, it was clear that students did not write details of how they arrived at their answers or conclusions. Few wrote more than one or two general statements. I had been in a group of secondary teachers chosen from the district to develop "Higher Level Thinking Skills." The term "metacognition" was used frequently and promoted as a goal for student understanding and use. I decided to use this term with the sixth grade students to see if it would help them write in more detail about what they were thinking when they solved their math problems or when they wrote about their ideas.

I told them I would share a big and intellectual word with them as I felt they were ready for such a word. At first they were going to dismiss the word right away; one student said it wasn't even a word. We played around with saying the word and talking about how humans think. I defined the word as "thinking about thinking."

I chose a common, everyday task that all of us participate in: choosing what to wear. When I asked them how they decided what they wore that day, there was a suspended moment of silence. It seemed a pretty dumb question. A few just said "they wore what they wore." This was a perfect beginning as it was the same kind of response they were giving as to how they were getting their math answers. I told them how I decided what I wore that day.

First I checked the weather in the paper and by looking outside. Several agreed that they did this as well.

I had to decide between wearing a dress/skirt or pants since either one could make a difference in accessories, in comfort, and in the impression I wish to give. Most of the girls related to this one.

I had to be sure what I wanted to wear was clean and ironed. They laughed at this one. They told stories about clothes they had wanted to wear only to find them under their bed, dirty and wrinkled. The ironing part puzzled them. They claimed no one ironed clothes any more.

I had to check to see if I had shoes that went with the possible outfit.

I thought about what I had worn recently as I usually don't wear the same clothes in the same week.

I thought about the color I felt like wearing.

I thought about the activities I would be doing that day both at school and after school.

Each new consideration brought out comments from them. Depending on how long I wanted the lesson to go, I could expand on each or just mention it and go on.

After this discussion, it was clear to students that their thinking process is richer than they first suspected. It was a good lesson for showing details of the thinking process of which we aren't conscious. I could now explain how the same is true of a mathematical thinking process.

Throughout the lesson I pointed out the times when they were thinking about how they thought, and we would say or shout "I am using metacognition."

For their journal entry they wrote about how they chose their outfit. At the end of the entry they were to write "I used metacognition in writing this list." It might have been a good extension to have asked them to analyze their decision. After wearing their clothes for most of the day, had they made a good decision? Why or why not? Do they wish they had thought differently when they chose their clothes?

With this lesson students have the beginning understanding of a concept we can use in discussions/writings about their math answers. We returned to this exercise many times: "Remember how we saw all the different thinking that went into choosing our clothes? Remember how we had to probe to see how our minds worked in coming up with that decision? Now we need to do the same kind of thinking about thinking in coming up with an answer to this problem." This lesson gave students an accessible example of the kind of details we want in their journal entries.

**Lesson Plan #2: Peer Evaluation of Journal Entries**

**Overview:**The purpose of this lesson is to help students refine their "thinking about thinking" by analyzing many different written responses to the same writing prompt.

**Strategy:**Compare student journal responses to the same math question. Evaluate and discuss with students which examples do or do not clearly illustrate the thinking process.

**Time Needed:**15-25 minutes, depending on amount of student interaction.

**Materials Needed**: Overhead projector; pens for the overhead; student writing journals; a copy of the original math question on a transparency; copies of four to eight student answers from their journals, each on a separate transparency (to give you a variety of answers to choose from during the lesson). Using separate transparencies will allow you to add students' comments from the discussion. Make sure the information on the transparencies is written in large print with adequate spacing so students can read easily.

**Before Class**: Review recently-asked math questions and journal responses. Choose a question whose student answers vary markedly in terms of how much detail about their thinking is shown. Select four to eight student answers, some of which show detail and some that are in the realm of "I just knew." Do

__not__put student names on the samples.

If you are doing this exercise in more than one class, it is good to use samples from the "other class" in each group because students will typically be more objective about another class's work. This means you will need at least two sample answer overheads.

**Procedure:**

1. Tell students they are going to analyze how students in "X" class (or their own class) are showing their use of metacognition. (Keep using this word. Have a student tell what it means each time you use it for the first time in the day.) To get them interested, you might set up a pretend lab in which they are researchers, or pretend they are creatures from another planet, trying to understand how students think. Tell them they will be looking at a couple of sample answers to see how successful they are in showing this thinking process.

2. Show the original question on the overhead. Leave it showing throughout the lesson.

3. Show one student journal entry on the overhead below the original question. Have someone read it out loud.

4. Ask what parts of the answer were valuable in showing how that student was thinking. Underline or circle such parts if you wish. Let students comment as thoroughly as possible. Ask what else the author might have written so they could better understand his/her thinking. Write these additions on the overhead.

5. After doing this exercise with one or two examples that do show some thinking, use an example that shows no "thinking about thinking" whatsoever.

6. Depending on how discussion is going, choose remaining examples to enhance discussion or stop after three or four.

7. At the end of discussion, have students write in their journals about the kind of answers that showed metacognition, to allow them to reflect on the lesson. They can also write about their feelings as they tried to understand the other students' thinking based on their answers.

**Personal Narrative:**

**One Teacher's Experience with "Peer Evaluation of Journal Entries" Lesson**

It is a challenge to get students to write details about their thinking. Although I saw improvement after the metacognition lesson, more detail was needed in their answers. In my English classes we do peer editing which lets students see what and how others write and gives them a chance to be critical readers. I varied this for the sixth grade math class. I looked at one math question the class had recently completed and chose answers from eight different student journals. Some showed good detail and explanation, some showed little or none.

Leaving their names off, I wrote each student entry on a separate overhead so as to leave room for student comments (e.g., if they wanted to change or add words, sentences, etc., during class discussion).

I told them to think of themselves as the researchers in this project. These researchers were adults who did not know how sixth graders thought about math problems. The researchers were interested in being better math teachers. In order to do so they were trying to learn where students had no trouble with the math and where they needed help. Students liked this role.

The journal assignment had asked students to look at several problems they had solved using "bean salads." They had created "salads" using different ratios of three kinds of beans. Students had worked in small groups and had bowls of actual dried beans that they could use while figuring out the ratios. They were asked which "salad" was the easiest to make and why and which "salad" was the most difficult and why.

We looked at an entry that had four sentences. It was one of the better answers. Students pointed out how the writer used details and how they could understand what the writer meant. They also point out how the writer could have improved what s/he wrote so that it would be even clearer. We all agreed it was an adequate answer.

The next entry was "Number 1 was easy because it just was." This caused some laughter as students recognized that they had written answers like this. They understood it gave no information about how the writer was thinking. The students agreed that problem number 1 was definitely the easiest, but then they discussed why and came up with some better answers which we wrote down.

Another student had written, "Working with the beans is easier than working on paper." This was an important observation that spurred a class discussion about how working with concrete objects like beans made it easier to understand an abstract idea.

Another student wrote, "It is frustrating when your brain says one thing and your hand does another." There were immediate nods of agreement on this one. Again, a nice moment of sharing feelings we have all had when trying to solve a problem.

Students were very interested in this lesson. Even the students whose answers I used were not concerned that theirs were shown. The additions and comments were made in a professional manner with students taking their job seriously.

This lesson showed students what others were writing. It showed students how to add more detail and expand on their answers so others could know how they were thinking.

At the end of the discussion have students write in their journals. Since all students had written on this journal question, they could copy their original answer and then write an improved version; they could write about the kind of answers discussed in class that showed metacognition; or they could copy one of the answers and add detail to it.

Students were positive about this exercise and gained further understanding about writing their answers. This kind of peer evaluation exercise should be done regularly. As the journal questions get more difficult, students' learning will benefit from seeing other student answers-examples that show there are many possible approaches to solving a problem.

__Writing Opportunities in Math Class__**How and When to Use Journal Writing**

1. When new material has been introduced.

Ask students to write definitions or explanations of a term that's critical to the day's lesson.

__Sample Direction__: "Explain in your own words the meaning of the term ____________."

2. When the class looks disengaged or confused.

Ask students to write an explanation of something you were doing or a term you used. Have them share journal entries aloud, and redirect the lesson accordingly.

__Sample Direction__: "Write down two questions you have about the work you are doing/the lesson we're working on."

3. When collaboration with fellow students is appropriate.

Have students form small groups and work together to solve a problem on paper. This will get them to talk to each other-to ask questions and give explanations-all with the common goal of solving the problem. 4. When teaching the value of revising their work.

Occasionally ask students to pick a journal entry and revise it. This helps emphasize that journal writing is an initial effort that can be rethought and improved upon-the end product is less important than the process.

__Sample Direction__: "Review the last three entries in your journal. Select one to revise." Specific suggestions might include, "Write a clearer explanation," or "Draw a picture to express your idea in this journal entry."

**Sample Journal Questions/Writing Prompts**

Writing prompts can take many different forms. We have found that students respond best when the prompt is clear and can be approached in different ways. We recommend prompts that do the following:

1. Pinpoint a confusing or easily misunderstood mathematical idea

__For Example__: "Do 0.2 and 0.020 equal the same fraction? Explain your answer."

Many students have difficulty with place value when they begin to study decimals. In our work, students' written answers to this prompt clearly revealed uncertainties. As one student responded: "The zeroes don't matter, so .2 equals .2." This student does not appear to have a good understanding of place value, having over generalized the "hint" to ignore certain zeroes. Another student drew two grids in response to this prompt. In one grid she colored in two rows of tenths and in the other grid she colored in two hundredths. She concluded that 0.2 was "way more" than 0.020. Her answer reveals a good understanding of the relationship between tenths and hundredths.

2. Can be solved using different strategies

__For Example__: "Allison's team won 8 out of 10 games. Jennifer's team won 15 out of 18 games. Whose team won a greater fraction of its games? Explain your answer."

Students used different strategies to approach this problem . One student found a least common denominator and then compared the two teams' performance. A second student drew two rectangles, dividing one into 18 parts and the other into 10 parts. He then colored in 15 and 8 parts, respectively, of each rectangle. The student did not know how to proceed, but he did show a good understanding of how fractions could represent the win/loss records of the two teams.

3. Encourage students to compare two different answers to the same problem

__For Example__:

"Who is correct? The problem: Which fraction is biggest? 1/3 or 2/5?" Jamar's solution: 2/5 is bigger because 15 is the LCD and 1/3 equals 5/15. 2/5 equals 6/15. So 2/5 is biggest. Bill's solution: I used the calculator. I made them decimals and then compared the decimals. For 1.3, I divided 1 into 3 and got 3.0. Then I divided 2 into 5 to get 2.2. 3.0 is bigger than 2.2, so 1/3 is biggest. This prompt encourages students to consider two very different approaches to the same problem: the least common denominator (LCD) method taught in the text and a calculator. The two offered solutions also model fairly clear explanations. In this case a correct and an incorrect solution are offered. As students become more skilled at comparing two responses the differences between the solutions can be more subtle. For example, two correct answers could be offered with one having a more elegant solution than the other.

__Managing Math Journals: Helpful Tips__1. Provide students with thin, inexpensive journals. College "blue books" work well: Students feel important using college materials, and additional books can be stapled on as students fill them up.

2. Keep journals in class. Collect math journals each day so as not to lose them.

3. Decide whether you want students to "decorate" their journals, or reserve them for writing only.

4. Decide on a system for identifying journal entries. Rather than having students take time to copy the writing prompt, have them number or date the entries.

5. Develop system for distributing and collecting journals each day. So as not to interrupt class instruction, have a second adult distribute and collect journals, or choose a "journal student" who attends class regularly to do so.

6. Use a timer for some journal assignments. This will help keep students writing. Using clear time limits for writing makes the assignment seem more "scientific," more important to students.

__Encouraging Students as They Write__1. Be patient. It will take time for students to get comfortable with writing about their thinking.

2. Tell students you understand how new and different this is for them. Remind them that there are no "wrong" answers in writing about thinking.

3. If students indicate they have no more to say:

- Read over what they've written. Ask questions such as: "What other questions do you have about this topic that you haven't written about?" or "What's another way this could have been said?"

- Have them copy what they've written, so they'll get the idea that they are to write for the whole time given. (Often, they'll get bored with copying and begin writing something new.)

- Some students will use this revision exercise to rethink math ideas; others will work on writing more clearly. Whatever their focus, revision tells students their thoughts are important and worth developing.

- Let students know you took time to read their journals.

- You won't have time for in-depth comments on each journal for every assignment, so try other kinds of feedback, too:

- Put stars by sentences that helped you see their thinking.

- Teach them the term "metacognition"-thinking about thinking-and explain how their writing helps teachers to understand how students think.

- Other purposes of writing: Writing is a concrete way to show students' thinking that they can look at and think about. Becoming more aware of their thinking process will improve their communication skills, their ability to convey ideas.

- Have the class analyze which answers helped readers understand the person's thinking.

__Mathography Prompts__Here are some ideas for prompts to get students to write a "mathography"-a sort of autobiography of their history with mathematics:

1. Write down some of the early math accomplishments that you remember from when you were little. For instance, when and how did you learn to count? How old were you when you could first count to one hundred? Who taught you? How did they teach you? Did you "show off" this new talent to others?

2. When you were in first, second, or third grade what did you like about math? What didn't you like about math at that time?

3. What do you remember about learning to add and to subtract? Which did you think was more fun? Why did you like that one better?

4. What was your teacher's name in first, second, or third grade? _______________ What kind of teacher was he or she in regard to teaching mathematics?

5. Did you have any "tricks" you used to remember adding or subtracting?

6. In what ways is adding and subtracting important?

7. Was math ever your favorite subject? ______ If so, when was it? What about math made it your favorite? If math has never been your favorite subject, what about it do you not like?

8. From your experience, do you think boys or girls tend to like math better? What makes you think this?

9. Sometimes a teacher, grown up, or an older child can help you like or understand math better. Did that ever happen to you? If so, tell about it. If not, tell about how that would have made a difference for you.

10. Sometimes people can recognize a time when their opinion of math dramatically changed either for the better or the worse. If such a time happened for you or for a friend of yours, tell about it. If you did not experience such a thing, tell about your steady feelings about mathematics.

11. Lots of times students think what they learn in math is only for the classroom and is really not of much use outside math class. Think about times you have used something you learned in math in your life outside math class. List some of those times when you used math outside of school.

12. What year in school was math the best for you? ________ What made it a good year in terms of math?

13. What year in school was math one of the worst for you? ________ What made it a bad year in terms of math?

14. If you were in a lengthy conversation about math or math class with friends of yours, what would be some of the things you would say? What would be some of the things they would say?

15. Draw a picture of you and the idea of mathematics.

16. Draw a picture of all you know about mathematics.