From: Subject: Using Writing In Mathematic Date: Wed, 26 Sep 2007 13:20:56 -0400 MIME-Version: 1.0 Content-Type: multipart/related; type="text/html"; boundary="----=_NextPart_000_0000_01C80040.1264C500" X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.3138 This is a multi-part message in MIME format. ------=_NextPart_000_0000_01C80040.1264C500 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Content-Location: http://www2.ups.edu/community/tofu/lev2/journaling/writemath.htm Using Writing In Mathematic
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Using Writing In = Mathematics

This strand provides a developmental model for incorporating writing = into a=20 math class. The strand includes specific suggestions for managing = journals,=20 developing prompts for writing, and providing students with feedback on = their=20 writing. In addition, the site includes two sample lessons for = introducing=20 students to important ideas related to writing about their mathematical=20 thinking.

 

3D"What=20

Materials

paper and pencil

overhead pens

overhead projector

calculators (optional)

 

Teaching Strategies For=20 Incorporating Writing Into Math Class: Moving From Open-Ended Questions = To Math=20 Concepts

Starting Out Gently with Affective, Open-Ended Prompts

Writing about thinking is challenging. For this reason, it's best not = to=20 start out having students write about unfamiliar mathematical ideas. = First get=20 them used to writing in a math class:

Sample Direction #1: Reflect on your participation in = class=20 today and complete the following statements:

I learned that I...=20
I was surprised that I...=20
I noticed that I...=20
I discovered that I...=20
I was pleased that I...=20

Sample Direction #2: Describe how you feel about solving = _________ problem.

 

 

 =20
Encourage students to keep their pencils moving. =

 

Try requiring 20 words per answer, even if they have to copy the = same=20 words again to reach 20.=20
=20
Use a timer. Ask students to keep writing until they hear the = timer go=20 off, to encourage them to write for the entire time and to = discourage=20 clock-watchers. Start by giving them two minutes to write, and work = up to=20 five or ten minutes at a time.

 

Next Step: Getting Students to Write about Familiar Mathematical=20 Ideas

1. Once your students have become accustomed to writing about their = attitudes=20 and feelings toward mathematics in their journals, they are ready to = write about=20 simple, familiar math concepts. It is important not to make the writing = too=20 difficult by asking them to write about unfamiliar math ideas. Using = writing to=20 review familiar math ideas will increase confidence and skill in writing = as well=20 as revisit important math concepts.

Sample Directions:

Explain in your own words what subtraction means.=20
Explain what is most important to understand about = fractions.=20

 

2. Use student writing samples to help them refine their writing. = (Note: Let=20 them write for a while before discussing examples, so their initial = ideas will=20 be their own.)

3. Introduce the term metacognition to help students understand the = reason=20 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=20 mathematical ideas, including concepts being taught at their current = grade=20 level. To help you move your students into this more advanced level of = writing=20 about their thinking. Here are some other suggestions to help you:

1. Encourage your students to use drawings and graphs to explain = their=20 thinking.

2. As student writing progresses, ask students to write about their = small=20 group work.

 

 

 

Getting Started with Two = Lesson=20 Plans

Lesson Plan #1: "Metacognition"- = Teaching=20 Students to Think About Their Thinking

Overview: The purpose of this lesson is to help students begin = to=20 understand how to communicate their thinking.

Strategy: Introduce the term "metacognition" and lead students = through=20 exercises that illustrate the concept of "thinking about thinking."

Time Needed: 15-25 minutes, depending on amount of student=20 interaction/participation.

Materials Needed: Overhead projector, pens for overhead, = writing=20 journals

Procedure:

1. Introduce "metacognition" by writing it on the overhead and = explaining=20 what it means. Get students to say the word aloud a few times to make it = less=20 intimidating. Then you might tell a story about research in how the = human brain=20 works.

2. Tell students that the more attention we pay to our thinking, the = more=20 we'll come to understand about the process of thinking. Although we're = used to=20 just being concerned about the results or the "answers," if we pay more=20 attention to how we think, it would help us to think more clearly, and = improve=20 the quality of our results.

3. As a non-threatening exercise to illustrate metacognition, ask = students=20 how they decided what to wear that day. Ask three or four students to = share=20 their answers out loud. Students will most likely give simple answers = like, "I=20 just wore what I wore," unaware of their unconscious decision-making=20 process.

4. If no students mention weather, style issues, etc., as part of = their=20 thought process, give prompts to stimulate discussion such as: "Did you = consider=20 style, weather, what your friends might wear, what you wore yesterday?" = Ask=20 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=20 considerations for choosing clothing that day.

Teacher Note: Point out several times during the = discussion that=20 the students are using metacognition-that they are thinking about = their=20 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=20 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=20 their choice in clothes for the day. Had they made the best choice? List = reasons=20 why/why not and relate to their original list of considerations. Do they = wish=20 they had thought differently when choosing their clothes for this = day?

End Result of Lesson: Students will have a beginning = concept to=20 use in their discussions/writings about their math answers; and = you, as a=20 teacher, have the clothing example to return to many times as an = example=20 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=20 not write details of how they arrived at their answers or = conclusions. Few=20 wrote more than one or two general statements. I had been in a = group of=20 secondary teachers chosen from the district to develop "Higher = Level=20 Thinking Skills." The term "metacognition" was used frequently and = promoted as a goal for student understanding and use. I decided to = use=20 this term with the sixth grade students to see if it would help = them write=20 in more detail about what they were thinking when they solved = their math=20 problems or when they wrote about their ideas.

I told them I would share a big and intellectual word with them = as I=20 felt they were ready for such a word. At first they were going to = dismiss=20 the word right away; one student said it wasn't even a word. We = played=20 around with saying the word and talking about how humans think. I = defined=20 the word as "thinking about thinking."

I chose a common, everyday task that all of us participate in: = choosing=20 what to wear. When I asked them how they decided what they wore = that day,=20 there was a suspended moment of silence. It seemed a pretty dumb = question.=20 A few just said "they wore what they wore." This was a perfect = beginning=20 as it was the same kind of response they were giving as to how = they were=20 getting their math answers. I told them how I decided what I wore = that=20 day.

First I checked the weather in the paper and by looking = outside.=20 Several agreed that they did this as well.

I had to decide between wearing a dress/skirt or pants since = either one=20 could make a difference in accessories, in comfort, and in the = impression=20 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=20 laughed at this one. They told stories about clothes they had = wanted to=20 wear only to find them under their bed, dirty and wrinkled. The = ironing=20 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=20 outfit.

I thought about what I had worn recently as I usually don't = wear the=20 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=20 and after school.

Each new consideration brought out comments from them. = Depending on how=20 long I wanted the lesson to go, I could expand on each or just = mention it=20 and go on.

After this discussion, it was clear to students that their = thinking=20 process is richer than they first suspected. It was a good lesson = for=20 showing details of the thinking process of which we aren't = conscious. I=20 could now explain how the same is true of a mathematical thinking=20 process.

Throughout the lesson I pointed out the times when they were = thinking=20 about how they thought, and we would say or shout "I am using=20 metacognition."

For their journal entry they wrote about how they chose their = outfit.=20 At the end of the entry they were to write "I used metacognition = in=20 writing this list." It might have been a good extension to have = asked them=20 to analyze their decision. After wearing their clothes for most of = the=20 day, had they made a good decision? Why or why not? Do they wish = they had=20 thought differently when they chose their clothes?

With this lesson students have the beginning understanding of a = concept=20 we can use in discussions/writings about their math answers. We = returned=20 to this exercise many times: "Remember how we saw all the = different=20 thinking that went into choosing our clothes? Remember how we had = to probe=20 to see how our minds worked in coming up with that decision? Now = we need=20 to do the same kind of thinking about thinking in coming up with = an answer=20 to this problem." This lesson gave students an accessible example = of the=20 kind of details we want in their journal=20 entries.

 

Lesson Plan #2: Peer Evaluation of Journal=20 Entries

Overview: The purpose of this lesson is to help students = refine their=20 "thinking about thinking" by analyzing many different written responses = to the=20 same writing prompt.

Strategy: Compare student journal responses to the same math = question.=20 Evaluate and discuss with students which examples do or do not clearly=20 illustrate the thinking process.

Time Needed: 15-25 minutes, depending on amount of student=20 interaction.

Materials Needed: Overhead projector; pens for the overhead; = student=20 writing journals; a copy of the original math question on a = transparency; copies=20 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=20 from the discussion. Make sure the information on the transparencies is = written=20 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=20 much detail about their thinking is shown. Select four to eight student = answers,=20 some of which show detail and some that are in the realm of "I just = knew." Do=20 not put student names on the samples.

If you are doing this exercise in more than one class, it is good to = use=20 samples from the "other class" in each group because students will = typically be=20 more objective about another class's work. This means you will need at = least two=20 sample answer overheads.

Procedure:

1. Tell students they are going to analyze how students in "X" class = (or=20 their own class) are showing their use of metacognition. (Keep using = this word.=20 Have a student tell what it means each time you use it for the first = time in the=20 day.) To get them interested, you might set up a pretend lab in which = they are=20 researchers, or pretend they are creatures from another planet, trying = to=20 understand how students think. Tell them they will be looking at a = couple of=20 sample answers to see how successful they are in showing this thinking=20 process.

2. Show the original question on the overhead. Leave it showing = throughout=20 the lesson.

3. Show one student journal entry on the overhead below the original=20 question. Have someone read it out loud.

4. Ask what parts of the answer were valuable in showing how that = student was=20 thinking. Underline or circle such parts if you wish. Let students = comment as=20 thoroughly as possible. Ask what else the author might have written so = they=20 could better understand his/her thinking. Write these additions on the=20 overhead.

5. After doing this exercise with one or two examples that do show = some=20 thinking, use an example that shows no "thinking about thinking" = whatsoever.

6. Depending on how discussion is going, choose remaining examples to = enhance=20 discussion or stop after three or four.

7. At the end of discussion, have students write in their journals = about the=20 kind of answers that showed metacognition, to allow them to reflect on = the=20 lesson. They can also write about their feelings as they tried to = understand the=20 other students' thinking based on their answers.

Personal Narrative:=20

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

It is a challenge to get students to write details about their=20 thinking. Although I saw improvement after the metacognition = lesson, more=20 detail was needed in their answers. In my English classes we do = peer=20 editing which lets students see what and how others write and = gives them a=20 chance to be critical readers. I varied this for the sixth grade = math=20 class. I looked at one math question the class had recently = completed and=20 chose answers from eight different student journals. Some showed = good=20 detail and explanation, some showed little or none.

Leaving their names off, I wrote each student entry on a = separate=20 overhead so as to leave room for student comments (e.g., if they = wanted to=20 change or add words, sentences, etc., during class = discussion).

I told them to think of themselves as the researchers in this = project.=20 These researchers were adults who did not know how sixth graders = thought=20 about math problems. The researchers were interested in being = better math=20 teachers. In order to do so they were trying to learn where = students had=20 no trouble with the math and where they needed help. Students = liked this=20 role.

The journal assignment had asked students to look at several = problems=20 they had solved using "bean salads." They had created "salads" = using=20 different ratios of three kinds of beans. Students had worked in = small=20 groups and had bowls of actual dried beans that they could use = while=20 figuring out the ratios. They were asked which "salad" was the = easiest to=20 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=20 answers. Students pointed out how the writer used details and how = they=20 could understand what the writer meant. They also point out how = the writer=20 could have improved what s/he wrote so that it would be even = clearer. We=20 all agreed it was an adequate answer.

The next entry was "Number 1 was easy because it just was." = This caused=20 some laughter as students recognized that they had written answers = like=20 this. They understood it gave no information about how the writer = was=20 thinking. The students agreed that problem number 1 was definitely = the=20 easiest, but then they discussed why and came up with some better = answers=20 which we wrote down.

Another student had written, "Working with the beans is easier = than=20 working on paper." This was an important observation that spurred = a class=20 discussion about how working with concrete objects like beans made = it=20 easier to understand an abstract idea.

Another student wrote, "It is frustrating when your brain says = one=20 thing and your hand does another." There were immediate nods of = agreement=20 on this one. Again, a nice moment of sharing feelings we have all = had when=20 trying to solve a problem.

Students were very interested in this lesson. Even the students = whose=20 answers I used were not concerned that theirs were shown. The = additions=20 and comments were made in a professional manner with students = taking their=20 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=20 could know how they were thinking.

At the end of the discussion have students write in their = journals.=20 Since all students had written on this journal question, they = could copy=20 their original answer and then write an improved version; they = could write=20 about the kind of answers discussed in class that showed = metacognition; or=20 they could copy one of the answers and add detail to it.

Students were positive about this exercise and gained further=20 understanding about writing their answers. This kind of peer = evaluation=20 exercise should be done regularly. As the journal questions get = more=20 difficult, students' learning will benefit from seeing other = student=20 answers-examples that show there are many possible approaches to = solving a=20 problem.

 

Writing Opportunities in = Math=20 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=20 critical to the day's lesson.

Sample Direction: "Explain in your own words the meaning = of the=20 term ____________."

 

2. When the class looks disengaged or confused.

Ask students to write an explanation of something you were doing = or a=20 term you used. Have them share journal entries aloud, and redirect = the=20 lesson accordingly.

Sample Direction: "Write down two questions you have = about the=20 work you are doing/the lesson we're working=20 on."

 

3. When collaboration with fellow students is appropriate.

Have students form small groups and work together to solve a = problem on=20 paper. This will get them to talk to each other-to ask questions and = give=20 explanations-all with the common goal of solving the problem.=20

4. When teaching the value of revising their work.

Occasionally ask students to pick a journal entry and revise it. = This=20 helps emphasize that journal writing is an initial effort that can = be=20 rethought and improved upon-the end product is less important than = the=20 process.

Sample Direction: "Review the last three entries in your = journal. Select one to revise." Specific suggestions might = include, "Write=20 a clearer explanation," or "Draw a picture to express your idea in = this=20 journal entry."

 

Sample Journal Questions/Writing Prompts

Writing prompts can take many different forms. We have found that = students=20 respond best when the prompt is clear and can be approached in different = ways.=20 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=20 your answer."

 

Many students have difficulty with place value when they begin to = study=20 decimals. In our work, students' written answers to this prompt clearly = revealed=20 uncertainties. As one student responded: "The zeroes don't matter, so .2 = equals=20 .2." This student does not appear to have a good understanding of place = value,=20 having over generalized the "hint" to ignore certain zeroes. Another = student=20 drew two grids in response to this prompt. In one grid she colored in = two rows=20 of tenths and in the other grid she colored in two hundredths. She = concluded=20 that 0.2 was "way more" than 0.020. Her answer reveals a good = understanding of=20 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=20 team won 15 out of 18 games. Whose team won a greater fraction of = its=20 games? Explain your = answer."

 

Students used different strategies to approach this problem . One = student=20 found a least common denominator and then compared the two teams' = performance. A=20 second student drew two rectangles, dividing one into 18 parts and the = other=20 into 10 parts. He then colored in 15 and 8 parts, respectively, of each=20 rectangle. The student did not know how to proceed, but he did show a = good=20 understanding of how fractions could represent the win/loss records of = the two=20 teams.

3. Encourage students to compare two different answers to the same=20 problem

For Example:

"Who is correct? The problem: Which fraction is biggest? 1/3 = or=20 2/5?"=20
=20
Jamar's solution: 2/5 is bigger because 15 is the LCD and = 1/3 equals=20 5/15. 2/5 equals 6/15. So 2/5 is biggest.=20
=20
Bill's solution: I used the calculator. I made them decimals = and=20 then compared the decimals. For 1.3, I divided 1 into 3 and got = 3.0.=20 Then I divided 2 into 5 to get 2.2. 3.0 is bigger than 2.2, so = 1/3 is=20 biggest.

 

This prompt encourages students to consider two very different = approaches to=20 the same problem: the least common denominator (LCD) method taught in = the text=20 and a calculator. The two offered solutions also model fairly clear=20 explanations. In this case a correct and an incorrect solution are = offered. As=20 students become more skilled at comparing two responses the differences = between=20 the solutions can be more subtle. For example, two correct answers could = be=20 offered with one having a more elegant solution than the other.

 

Managing Math Journals: = Helpful=20 Tips

1. Provide students with thin, inexpensive journals. College "blue = books"=20 work well: Students feel important using college materials, and = additional books=20 can be stapled on as students fill them up.

2. Keep journals in class. Collect math journals each day so as not = to lose=20 them.

3. Decide whether you want students to "decorate" their journals, or = reserve=20 them for writing only.

4. Decide on a system for identifying journal entries. Rather than = having=20 students take time to copy the writing prompt, have them number or date = the=20 entries.

5. Develop system for distributing and collecting journals each day. = So as=20 not to interrupt class instruction, have a second adult distribute and = collect=20 journals, or choose a "journal student" who attends class regularly to = do=20 so.

6. Use a timer for some journal assignments. This will help keep = students=20 writing. Using clear time limits for writing makes the assignment seem = more=20 "scientific," more important to students.

 

Encouraging Students = as They=20 Write

1. Be patient. It will take time for students to get comfortable with = writing=20 about their thinking.

2. Tell students you understand how new and different this is for = them.=20 Remind them that there are no "wrong" answers in writing about = thinking.

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

 

 

 

4. Ask students to choose a journal entry to revise.

 

 

5. Provide feedback.

 

 

 

Stars=20
=20
Stickers=20
=20
Smiley Faces

 

 

6. Clearly communicate to students the purpose of writing in math = class.

 

 

 

7. Use students' writing samples to help them refine their = writing.

 

 

Mathography=20 Prompts

Here are some ideas for prompts to get students to write a = "mathography"-a=20 sort of autobiography of their history with mathematics:

1. Write down some of the early math accomplishments that you = remember from=20 when you were little. For instance, when and how did you learn to count? = How old=20 were you when you could first count to one hundred? Who taught you? How = did they=20 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=20 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=20 think was more fun? Why did you like that one better?

4. What was your teacher's name in first, second, or third grade?=20 _______________ 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=20 math made it your favorite? If math has never been your favorite = subject, what=20 about it do you not like?

8. From your experience, do you think boys or girls tend to like math = better?=20 What makes you think this?

9. Sometimes a teacher, grown up, or an older child can help you like = or=20 understand math better. Did that ever happen to you? If so, tell about = it. If=20 not, tell about how that would have made a difference for you.

10. Sometimes people can recognize a time when their opinion of math=20 dramatically changed either for the better or the worse. If such a time = happened=20 for you or for a friend of yours, tell about it. If you did not = experience such=20 a thing, tell about your steady feelings about mathematics.

11. Lots of times students think what they learn in math is only for = the=20 classroom and is really not of much use outside math class. Think about = times=20 you have used something you learned in math in your life outside math = class.=20 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=20 good year in terms of math?

13. What year in school was math one of the worst for you? ________ = What made=20 it a bad year in terms of math?

14. If you were in a lengthy conversation about math or math class = with=20 friends of yours, what would be some of the things you would say? What = would be=20 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.

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