Topics: Introduction CLAD Goals Course Materials Course Assignments Semester Schedule Assignments: Mathematics-Language Arts Trade Book Student Interview Sample Interview Peer Teacher Mathematics Unit Plan Assignments Assignment 1 Assignment 2 Assignment 3 Assignment 4 Assignment 5 Back to Syllabi Table of Contents

Introduction:

Learning to teach mathematics well is difficult, and this course will not complete your education in learning how to teach mathematics. Rather, this course is but one stage in what I hope will be a continuing evolution of you as a mathematics teacher. By the end of the course, it is my hope that you will begin to see yourself as an intelligent consumer of mathematics education and will have developed the ability to ask the important questions that will point you toward the creation of a rich mathematical environment in your classroom.

1. Widening the repertoire of communication strategies and skills.
   • Learn to orchestrate classroom discourse in ways that promote the investigation and growth of mathematical ideas.
2. Creating a collaborative learning community.
   • Learn to create an environment that is safe for students to share and justify their thinking, ask questions, make conjectures, and take risks.
   • Realize the importance of positive student attitudes toward mathematics.
     
   • Learn to guide individual, small-group, and whole-class work.
3. Promoting teaching/learning strategies which align with constructivist theories.
   • Come to view mathematics as a sense-making activity.
     
   • Assess students mathematical thinking and their understanding of mathematics concepts and procedures, and plan instruction based on those understandings.
     
   • Respect your students thinking, even when you do not understand it.
4. Promoting higher order thinking skills.
   • Utilize methods which convey understanding of concepts and facts, not just rote memorization.
5. Developing greater sensitivity to and respect for cultural differences.
   • Be aware of issues regarding equity in teaching and learning mathematics.
     
   • Develop an understanding for the role that convention plays in mathematics.
6. Widening schema of educational environment.
   • Recognize the role that family and friends play in the mathematical development of children.
7. Understanding global linkages/interconnections within and among personal, social environments, technological systems.
   • Develop an understanding for the role technology might play in mathematics education.
8. Promoting understandings of interconnections among content areas through thematic, interdisciplinary instruction.
   • Come to see mathematics in all aspects of our daily lives and make mathematics a subject throughout the school day, not just during mathematics.
     
   • Use literature as a way to engage students in mathematical thinking.
9. Developing English language skills across all content areas while supporting/respecting childrens primary language.
   • Encourage students to explain their mathematical reasoning in their native languages as well as in English.
10. Understanding the theoretical underpinnings of instructional practices.
    • Be aware of the current national and state recommendations regarding the teaching of elementary school mathematics, and plan instruction that follows these recommendations.
      
    • Develop a thorough understanding of the scope and sequence of the elementary school mathematics curriculum.
      
    • Be familiar with a wide variety of activities and manipulative materials for use in mathematics instruction.
11. Inspiring a life-long commitment to curiosity and learning.
    • Join NCTM and GSDMC.
      
    • Become an intelligent consumer of mathematics education.
      
    • Developed the ability to ask the important questions which will guide you in developing the mathematics curriculum for your students.
      
    • Become aware of your own weaknesses and develop strategies for dealing with them.
      
    • Come to see learning to teach as a process that extends throughout your professional career.

Required Course Materials:

Elementary School Mathematics: Teaching Developmentally, by John Van de Walle

Packet of xeroxed materials (labeled "Philipp, TE 910A, Teaching Mathematics in the Elementary School"), available at Aztec Shops Copy Center

Join NCTM and GSDMC and attend the 1998 Annual GSDMC meeting in San Diego, Feb 1998

Strong Suggestion: Join CMC

Optional Course Materials:

Mathematics Framework for California Public Schools, K-12, California Department of Education (1992)

• Fraction Circle Activities by B. Berman et al.
  
• A Collection of Math Lessons, Grades 1 - 3, by M. Burns et al.
  
• A Collection of Math Lessons, Grades 3 - 6, by M. Burns

Math for Girls and Other Problem Solvers by D. Downie et al.

IDEAS from the Arithmetic Teacher--Grades 4-6, F. Fennell, Ed.

IDEAS from the Arithmetic Teacher--Grades 1-4, G. Immerzeel, Ed.

Base Ten Mathematics, by M. Laycock

Skateboard Practice: Multiplication and Division, by P. McLean et al.

How to Develop Problem Solving Using a Calculator, by J. Morris

Curriculum and Evaluation Standards for School Mathematics, National Council of Teachers of Mathematics (1989)

Professional Teaching Standards for Teaching Mathematics, National Council of Teachers of Mathematics (1991)

Skateboard Practice: Addition and Subtraction, by Robertson et al.

• Fractions with Pattern Blocks, by Zullie

Non-graded assignments

Peer teaching

GSDMC conference attendance and reflection

Mathematics learning kit

Student Interview Write-ups

Semester Schedule

This schedule will change as I better understand your needs. This is the first time I have used the videoclips, so I expect you to help me think about how to make their use relevant for you. The topics listed are to provide a general idea of what might be covered, but I never end up covering all of the them in this order. We will make adjustments as the course progresses.)

Week 1:  Jamacha
#1	Wed Sept 3	Introduction
        8:30-11:30	The syllabus and the semester
	                Star Activity	
                        The History of Mathematics Education

#2	Fri Sept 5	How People Learn  (Constructivism)
	8:30-11:30	Video Clip #1
			Homework #1 Due	


Week 2:  Jamacha
#3	Mon Sept 8	Sample Interview  
	8:30-11:30	Early Number
			Addition and Subtraction Problem Types and Solution 
                        Strategies
			Videoclip #2

#4	Tues Sept 9	Addition and Subtraction Problem Types and 
        8:30-11:30      Solution Strategies 
	        	Interview 1  (Grade 2)
			Homework #2 Due
			Videoclip #3

#5	Thur Sept 11	Place Value
	8:30-11:30	Interview 1 write-up due (2 copies)
			Videoclips #4 & #5


Week 3:  Jamacha
#6	Tues Sept 16	Multiplication and Division Problems
	8:30-11:30	Interview 2 (3rd Grade)
			Videoclips #6 & #7

#7	Thur Sept 18	Peer Teaching:  Lesson Plan Due
	8:30-11:30	Interview 2 write-up due (2 copies)
			Videoclip #8

 Week 4:  Jamacha
#8	Tues Sept 23	Place value, Multiplication, and Division(cont') 
	12:00-3:00	Videoclip #9

#9	Thur Sept 25	Joint Mathematics/Language Arts Lesson
	8:30-11:30	Outline of Unit Plan (Unit Plan Part a)
			Videoclip #10


Week 5:  Jamacha
#10	Tues Sept 30	Algorithms/Basic Facts
	8:30-11:30	Math/Lang Arts Presentation
			Videoclip #11

#11	Wed Oct 1	Interview 3  (5th Grade)
	8:30-11:30	Fractions
			Homework #3 Due 
			Videoclip #12			

Week 6:  Jamacha
#12	Mon Oct 6	Fractions
	8:30-11:30	Pre-Assessment Due (Unit Plan Part b)
			Interview 3 write-up due (2 copies)
			Videoclip 13


Week 7:  Jamacha
#13	Mon Oct 13	Fractions/Decimals
	8:30-11:30	Homework #4 Due (Fractions)


#14	Thur Oct 16	Place Value and Other Bases (Janet Bowers)
	8:30-11:30	Unit Plan Due
			

Week 8:  Jamacha
#15	Mon Oct 20	Conceptual and Calculational Orientations
	12:00-3:00	Equity
			Prealgebra/Algebra	
			Exam

Purpose: The purpose of this assignment follows:
Design a set of questions to accompany a trade book appropriate for primary grade students. The questions should serve the following two purposes:

1. To be used to facilitate the students listening skills, including interpretive and predictive skills. These questions will be created in conjunction with your language arts instructor
   
2. To be used as the basis of a mathematics lesson. The context for these questions ought to relate the plot of the story. These questions will be created in conjunction with your mathematics methods instructor.

Assignment:

• Select a trade book that might be of interest to children in the second grade. (You may consider a different grade if you prefer.)
  
• Prepare questions you might ask your students as you read this book. You may want to insert post-its throughout the book so you do not forget when to pose the appropriate questions. Questions ought to focus the children on what has happened in the story, what they think or feel about what has happened, and what might happen. Following are some examples:
  • Who is visiting PJ Funnybunny?
    
  • Why are PJ Funnybunnys parents angry with him?
    
  • How do you think PJ Funnybunny feels about what his cousin did?
    
  • What do you think will happen next?
    
  • What would you do if you were PJ Funnybunny?
• Prepare questions that might be used as the basis of a mathematics lesson. (Actually, these questions might be the basis for several mathematics lessons.) The questions ought to reflect various types of mathematics problems that second graders can solve. Be sure to include a variety of problem types and be prepared to use different sized numbers to respond to various students comfort levels. The mathematics questions might be posed while you are reading the book, or you may want to save most of the questions to be used in a separate mathematics lesson. In the event that you choose the latter, I suggest you still try to incorporate some mathematical questions into the reading of the book. Following are some examples:
  • This book has 42 pages. I am on page 22. How many more pages are there?
    
  • If PJ and his family were eating dinner, how many eyes would be at the table? How many arms? Legs? Fingers? Toes?
    
  • If PJ and his cousin had 12 apples to share, how many should they each get?
• You will have 25 minutes to present your book to a small group of second grade students and to your peers. During the presentation, you should read the book, posing the questions you have prepared. You should share at least one of the mathematics questions with the children, providing them with whatever tools you think they may need to help them. Your peers will observe the lesson and help you analyze how the experience went.
  
• Submit a write-up of this assignment, including the full reference for the trade book, the questions asked (along with the location in the story when you would ask them), and a brief overview explaining what you are hoping to accomplish with the questions. Provide enough detail so that other students could use this lesson. Bring one copy of your lesson for each student teacher in your group. (Also, have one for your Language Arts instructor and one for you Mathematics Methods Instructor).

To remember during the interview:

Set a friendly tone

Pay close attention to child (How s/he feels - what s/he is thinking - what s/he is doing)

Take copious notes on both child and interviewer

Consider wait time (You may be uncomfortable with silence, but the child may not be)

Don't assume child wants question repeated - ask

Purposes:

The purpose of this assignment is to provide you an opportunity to reflect on what you learned while interviewing an elementary school student about his or her knowledge within a specific mathematical content area.

Assignment:

Working in pairs, construct interview questions you intend to use during your interview. Arrangements will be made for you to conduct the interview during class time. After conducting the interview, write a review of the interview you conducted. The review should contain the following information (You and your partner may either submit separate reports, or you may work together and submit one report):

1. General Information
   • Your name and the name of your partner
     
   • The name, age, and grade of the student you interviewed
     
   • The teachers name
     
   • Any pertinent information about the child you would like to mention
2. Your analysis include:
   • All the questions you asked
     
   • What did you learn from this experience?
     
   • What impressions did you come away with from this experience?
     
   • Did anything surprise you?
     
   • What, if any, implications does this interviewing have for instruction?
     
   • How did you feel during the interview?

(Note: Avoid evaluative statements about the child, such as, "she was really smart" or "he seemed slow." You do not know enough about the child to make such statements, and besides, those statements do not provide any information. Instead, provide details, such as, "When I asked her what 8+9 was, she solved it by saying 8 and 8 is 16, and one more is 17. I thought that was neat because I would not have expected a child to do that," or "I asked him this question and he just looked at me. I asked him if I should repeat the question, and he said no. I did not know how else to reach him.")

1. Basics
   • Set child at ease: Ask childs name, and introduce yourself. etc...
     
   • Explain this is an assignment for your class. Tell child this will not be graded.
     
   • Some questions will be easy, some hard. You are not expected to get them all right. Thats OK. Just do best you can.
     
   • You are interested in how the child thinks, not how many correct answers they come up with. Throughout, you will be asking "how did you get this," or "what were you thinking." Do best you can.
     
   • Introduce materials: Counters, paper and pencil, fingers. Use whatever you are comfortable with.
     
   • Explain why you want to tape the interview and ask for permission
     
   • Initiate conversation by asking childs age, brothers, sisters, favorite things to do, etc...

     Remember: Set a friendly tone Pay close attention to child (How s/he feels - what s/he is thinking - what s/he is doing) Take copious notes on both child and interviewer Consider wait time (You may be uncomfortable with silence, but the child may not be) Don't assume child wants question repeated - ask Do not feel you must pose every question. If a student struggles with some questions, it would make sense to skip more difficult questions. Note, however, the questions are not in order of difficulty. Three factors that affect the difficulty students have with questions involve where the unknown is located, whether or not the problem involves action, and the size of the numbers.

2. The Interview
   1. How high can you count?
      
   2. What number comes after 8? After 19? After 27? After 73? After 100?
      
   3. Can you count by 2s? 3s? 5s? 10s? 100s?
      Addition and Subtraction Word Problems: Ill start with an easier one.
      
   4. JRU: Eric has 5 apples. His mom gave him 3 more apples. How many does he have altogether? (Will he directly model with counters? With fingers? Will he count on? Derived facts?)
      
   5. JRU: Eric has 4 toy cars. His friend gave him 7 more toy cars. How many toy cars does Eric have now? (If counting, does he count on from the smaller, or the larger?
      
   6. SRU: [ ] has 14 toy cars. He gives 5 toy cars to Eric. How many toy cars does [ ] have left? (I might use smaller numbers here, if information gathered from the interview indicates to me that these numbers are too difficult.)
      
   7. JCU: [ ] had six marbles. How many more marbles does he need to buy to have 13 altogether?
      
   8. PPW-WU: There are 8 boys and 5 girls in the room. How many children are in the room altogether?
      
   9. PPW-PU: Eric has 14 colored marbles. 8 are blue and the rest are red. How many red marbles does Eric have?
      
   10. JSU: [ ] has some toy cars. He goes to the store and buys 4 more toy cars, and then he has 9 toy cars. How many toy cars did [ ] have to start with?
       
   11. SSU: Eric had some apples. He gave 3 apples to [ ], and then he had 4 apples left. How many apples did Eric have before he gave 3 apples to [ ]?
       
   12. Compare Difference Unknown: Eric has 9 marbles and [ ] has 4 marbles. How many more marbles does [ ] have than Eric?
       
   13. Mult: A pack of gum has 5 pieces. How many pieces of gum would you have altogether if you had 3 packs of gum?
       
   14. Partitive Division: At a party, there were 18 M&MS left to be shared fairly among 3 children. How many M&MS would each child get?
       
   15. Measurement Division: 20 children are to be driven to the park. If each car had seat belts for only 4 children, how many cars would be needed to drive all 20 children to the park?
       
   16. Can you write a problem with bigger numbers that you can solve?

       Place Value:
       

   17. Put out 3 loose cubes and 4 rods of unifix cubes with ten cubes on each rod. "Each of these rods has the same number of cubes." Hold up several rods next to each other to show that they are the same length. "There are ten cubes in each of these rods. Do you want to count the cubes in one of the rods to be sure that there are ten? OK so there are ten cubes in each rod; can you tell me how many cubes there are altogether counting all the cubes?" Sweep hand over entire collection of rods and cubes.
       
   18. Put 1 more ten rod down: "Watch what I do. Im putting ten more here. Now how many are there?"
       
   19. Monday Anna played Nentendo for 20 minutes before school and 10 minutes in the evening. How many minutes did she play Nentendo on Monday?
       
   20. How many tens are there in 32?
       
   21. (For children with place value knowledge, see if they can invent procedures for adding multidigit numbers. Example: Misha has 34 dollars. How many dollars does she have to earn to have 57 dollars?)
       
   22. There were 28 girls and 35 boys on the playground at recess. How many children were there on the playground at recess?
       
   23. Imagine that this is a cookie. Could you show me how you might share this with two people so that each person gets the same amount? (Point to 1 piece and ask what you would call this.)

       Same for 3 people. 4 people If successful, 5 people.
       

   24. Which of the following are triangles?

Purposes: The purposes of this assignment are as follows:

1. Design a mathematics lesson activity based on an idea from the Arithmetic Teacher, your methods text, or some other teacher resource,
   
2. Gain experience in writing a lesson plan for mathematics, and
   
3. Practice teaching a hands-on mathematics lesson.

Assignment:

Select and read an article from the Arithmetic Teacher or other resource for ideas on teaching a mathematics topic. You may use a resource you have already read for your unit plan assignment. Build a hands-on lesson around an idea for teaching a mathematical concept which you read about.

1. Your lesson plan should include the following: a) lesson objective(s), b) an introduction, c) development, and d) closure. Write at least two questions which might be used to determine the informal (or preinstructional, or intuitive) knowledge related to your topic which your students possess. Also, write at least five questions you might ask in order to assess your students' knowledge as the lesson unfolds.
   
2. You will need to: a) identify the grade level for which the lesson is designed, b) name the mathematics unit in which the lesson would be included, c) list and/or sketch the materials you will use, and d) think about how you structure your time.
   
3. Bring in 4 copies of your completed lesson plan, 1 for me and 3 for your peers (preferably typed). Do not turn in manipulative materials or games. Provide sketches and descriptions of these materials. Do not turn in the article or directions for making materials.
   
4. You will use the hands-on materials during your demonstration lesson in the 910A class. You will "teach" a group of your peers for a maximum of 15 minutes. Your peers will give you feedback on your lesson. They may also be interested in your source(s) of ideas, so bring bibliographical data.
   
5. While you teach your lesson, 3 students will be responsible for focusing on each of the following three areas:
   1. Your presence: How do you hold yourself. Do you maintain eye contact? How do you speak? (Too fast or too slow; clearly; do you face the students, or do you talk to the chalkboard; Do you use repeated phrases such as "um" or "ah?") Do you look confident? Do you look intimidating? Friendly? Timid? Scared?
      
   2. Your lesson development: Did you capture and hold the interest of your students? Were the students engaged in the lesson? Were they busy doing, or thinking, or both? Did the ideas in your lesson flow? Was the pacing reasonable? Did you try to cover too much or too little material during the lesson? What was it that the students were supposed to be able do? To understand?
      
   3. Your questions: (Record every question asked during the lesson) What, if any, questions were asked during the lesson, and by whom (any questions asked by students)? How many of the questions were rhetorical or managerial (Are you with me, Tim?; Class, you all understand, dont you?)? How many of the questions were low level questions (Susan, 1 and 1 is what?)? How many of the questions were high level questions (How many eighths are there in 1 1/3?)? Hoe many questions were asked to assess the students understanding (Why do you think 5/7 is smaller than 5/9?; Tom, please explain the thinking that led to that answer.)?

Some other thoughts about planning lessons you would teach to children:

Be conscious of safety, nutrition, and too much emphasis on extrinsic rewards. If you make your own materials, make them attractive and durable enough to be used with children. Think about whether the materials will lead children to understand mathematics or will actually distract them from doing so. Keep in mind your real purpose in teaching the lesson. Beware of lessons which are cute, but have no substance; lessons which guarantee success in getting right answers, but do not challenge; and lessons which are "fun", but have no mathematics educational value.

Note: When acting as an observer for someone else's lesson, you may find it helpful to think like a child. However, behave like an adult. This is not a lesson on classroom management.

Purpose: The purpose of this activity is for you to gain experience in designing a mathematics unit plan.

Steps to developing the plan:

1. Decide upon a mathematics content strand for which you will design your unit.
   
2. Decide upon the grade level at which the unit will be taught.
   
3. Build your background knowledge by reading about the content area. (See bibliography assignment.)
   
4. Examine the teacher's edition of a mathematics textbook at the grade level you chose. Look for the content strand you chose and note the material covered and the sequence in which it is presented. (If you have a question about the logic of the sequence or content covered, discuss it with one of the instructors.)
   
5. List at least five broad content goals to be developed through your unit.
   
6. List at least two skill goals to be developed through your unit.
   
7. List at least two affective goals to be developed through your unit.
   
8. Plan a unit that will last at least two weeks.
   
9. Use resources, such as Van de Walle, the optional course materials, the Arithmetic Teacher , resource books, children's library books, and textbook materials in order to find lesson and activity ideas. Create a list of resources.
   
10. Interview at least 4 students from the class in which you will teach this unit. These interviews should take no more than 5 minutes each. The purpose of the interview is to determine what knowledge the students have about the concepts you plan to teach and the concepts that are prerequisite for your unit.
    
11. Outline the lessons that you will teach in order to meet the goals you have stated for the unit. (Objectives will be more specific for individual lesson plans.) Carefully sequence and connect the lessons to provide optimal learning experiences.
    
12. List all materials to be used. Provide diagrams of games, manipulatives and any other materials which cannot be turned in with the unit plan. Include copies of any worksheets you will use and page numbers of text pages with descriptions of the pages.
    
13. Follow format "Writing the final draft" for turning in assignment.

AN INTEGRATED UNIT OF STUDY

Writing the unit will require several drafts before you can expect it will be ready to put into a final draft. The process of developing a sequence of lessons, considering the necessity of prerequisite skills, and narrowing the topic to insure children learn concepts is time-consuming Please plan your time accordingly.

Preparation for a Unit

There are several important steps teachers should follow in preparing a unit of work. Listed below are four guideposts which describe the steps that a teacher should follow in thinking about a unit of study. These steps precede the final writing of the unit.

Guidepost Number One: Content

Read to gain an overview of the subject matter to be covered in the unit of study. Read what the California Mathematics Framework and the NCTM Standards say about the topic you intend to teach. Read the material in your mathematics methods textbook which covers the content of your unit.

Guidepost Number Two: Students

Who are the students you will be teaching? What do they already know about the mathematics, both in terms of procedural knowledge and conceptual knowledge? Do they understand the prerequisite knowledge necessary before they can learn the new material? If not, what will you do? What vocabulary will you be introducing? Are there multicultural or multilingual issues that might affect your unit?

Guidepost Number Three: Goals

Think through the justifications for teaching this unit to children and the goals toward which you will strive throughout the unit. These goals should be specific to the unit under consideration and should cover the major content, attitudes, and skills you will want to develop. Dividing goals into 3 categories will strengthen your unit and sharpen your thinking.

1. Knowledge/content goals - - what concepts and facts do you want children to know?
   
2. Skills goals - - what skills will the children develop as a result of the unit?
   
3. Affective objectives - - what attitudes, values, and/or appreciations do you want children to acquire toward the unit content?

These major goals of the unit will later be reflected in specific objectives which are stated for the daily lesson plans.

Guidepost Number Four: Materials

Survey the teaching materials which are available. Make a preliminary list of textbooks and library books, films, filmstrips, recordings, video tapes, computer software, manipulatives, and other materials. It is necessary to determine the availability of materials before further planning takes place. Visit your school and school district materials center. Preview the materials to be sure they are accurate and suitable for the population you are teaching.

Guidepost Number Five: Learning Experiences

When planning a unit, use what you have learned in the other methods courses. Integrate reading, language arts, math, science, social studies, and fine arts whenever it is appropriate. When planning an activity, teach the skills which children need to use to be successful. For example:

• if you are having children read a section of the textbook, teach children HOW to read in that content area.
  
• if you have children work with new manipulatives, give free exploration time, discuss properties of manipulatives. Ask the students for suggestions on appropriate ways to handle the manipulatives.
  
• if you are using calculators and or computers, find out how students perceive the technology and what their prior experiences were like.
  • if you are referring to a map, teach children HOW to interpret that map.
    
  • if you have children work in cooperative groups, teach them how to do so.
    
  • if you are cooking, make it a learning experience in reading, measurement, and cooperation.

Think through the possible activities to be used in the unit, giving attention to the learning experience in three major phases:

1. Initiation: Identify interest-getting devices which will motivate the children. Plan activities which will provide for exploration of the area of study by the children. Try to set up real-life problems which will give you information about children's prior knowledge of the content.
   
2. Carrying on the working period: Think about possible activities which will make up the major working period of the unit. For example, you may want to have the children divide into committees to carry out their investigation, or you may want each child to make a notebook on the subject matter being studied.
   
3. Culmination: Plan ways that the unit can be tied together at the end. This could be done through problem solving activities, students discussing and writing cooperative group reports, etc. Also, think about how you will evaluate the growth and learning of the children. As you plan your evaluation, you may find that you need a formal assessment of the knowledge/content objectives at the end of the unit, that you can analyze and observe daily work products to evaluate the skills objectives, and that you need to listen and observe children's responses to evaluate the affective objectives.

Writing the Final Draft

When you have completed the four guideposts, organize your unit following an outline such as this one:

Outline for Writing an Integrated Teaching Unit

1. Introduction Include the following items:
   1. Justification for the unit
      
   2. Description of the population to be taught
      
   3. Major content of the unit
      
   4. Length of time (length of periods - number of days)
2. Unit Goals Organize into three categories:
   1. Knowledge/information/content
      
   2. Skills
      
   3. Attitudes/appreciations
3. Instructional Aids Make an annotated list of the following:
   1. Films, filmstrips, audio, video tapes, manipulatives, computer software
      
   2. Reference books
      
   3. Trade books to be read to and by the class
      
   4. Textbooks
      
   5. Maps, charts, etc.
4. Organization Choose one of the following formats:
   1. Daily lesson plans
      
   2. Weekly block chart form

      	Mon.	Tues.	Wed.	Thurs.	Fri.
      Lesson Objective Major activity: Needed materials and/or resources

5. Description of Major Activities: Include on-going projects, such as:
   1. Development of vocabulary/concepts (think about Lesh's model)
      
   2. Bulletin board/wall displays
      
   3. Reporting activities
      
   4. Committee work
      
   5. Compilation of a booklet
      If you did not write daily lesson plans in IV, explain in this section the activities mentioned in the block chart.
6. Evaluation Include formal and informal measures, such as
   1. Written test - include some test items
      
   2. Participation in class activities
      
   3. Daily work
      
   4. D. Observation for specific behaviors
7. Appendix
   1. Sample materials (specific idea pages)
      1. Sample charts, materials you will use in class.
         
      2. Sketches of manipulatives, measurement tools, etc.
         
      3. Sketches of bulletin boards.
   2. Annotated bibliography of what you have read in preparation.

A well-prepared unit is evidence that you are a teacher who is able to plan and organize effectively. To present your work in the most enhancing way, be sure to do the following:

1. Type using a consistent style.
   
2. Proofread for mechanical errors.
   
3. Put into a booklet format.

Reading Assignment
Philipp Packet: Multicultural Mathematics and Alternative Algorithms, by Philipp
Philipp Packet: International Comparisons (by James Stigler) Van de Walle Chapters 1, 2

1) Assignment:

Write no more than 250 words sharing your thoughts. You may want to write a poem, or provide sentence fragments, or draw a picture, or write a song, or write short sentences -- but try to convey your reactions to these readings. Questions to consider include: What surprised you? What upset you? What pleased you? What implications do you see for teaching? What would you still like to know?

1. Pick a theme that you think might be of interest to first graders and write 9 different addition and subtraction word problems around that theme. The word problems should be written so as to include the different problem types on the Problem Type Chart in your readings. The chart includes 11 different problem types, but you need not write a Compare Quantity Unknown problem or a Referent Unknown Problem.
   
2. For each of the following pairs of problem types, circle the problem type that you would expect to me more difficult for first graders and provide one line of explanation. If a pair of problem types are of no appreciable difference, then circle both.

   A)	JCU	SRU
   B)	JCU	JSU
   C)	SRU	JSU
   D)	PPW-PU	PPW-WU
   E)	JRU	PPW-WU
   F) Pick two of your own and explain which is harder.

   JRU	-Join Result Unknown
   JCU	-Join Change Unknown
   JSU	-Join Start Unknown
   SRU	- Separate Result Unknown
   SCU	- Separate Change Unknown
   SSU	- Separate Start Unknown
   CDU	- Compare, Difference Unknown
   PPW-PU	- Part-Part Whole, Part Unknown
   PPW-WU	- Part-Part Whole, Whole Unknown

3. Describe 4 different solution strategies a first grade child might use to solve the following problem:
   Tom has 6 marbles. He buys some more marbles, and then he has 13 marbles. How many marbles did he buy?
   
   
4. Write at least 1 question or idea based on your readings.
   1. State your question
      
   2. Describe what prompted this question.
      
   3. Discuss why it is important that the matters addressed by this question be discussed. Put another way, why is your question significant?

Mack, N. (1993) King Fredericks Fractions
Nancy Macks Critical Ideas

Read the King Fredericks Fractions story to a student or to a small group of students. Either prepare fraction circles, or have colored markers and paper to draw circles as you read. (It might be better to have the circles.) Write a reflection of your experience. (Note: I would look for an elementary school child in the 4th or 5th grade.)

Audiotape the experience, then listen to it and reflect upon the experience. Some of the questions you may want to consider:

• What did the child (or children) seem to understand? What didnt the child seem to understand?
  
• What seemed to help the child understand?
  
• What surprised you about the experience?
  
• Did you focus on what the child was doing, or on what the child was thinking? (Do you see the difference?)
  
• What role did you play? Did you ask questions, or were you more inclined to tell the child? What kinds of questions did you ask?

1. Write two division problems to represent the expression 32 ÷ 8. One of the problems you write should be a measurement division problem and the other should be a partitive division problem.
   • Explain how these two problems might be modeled differently by a child.
2. Sally read 8 books in 34 weeks. She wanted to figure out about how long she spent reading each book, and so she divided:
   Sally is confused about what the 2 represents.
   1. Consider the 2. Of what is it a number? (Or, put another way, "The 2 is a number of what?")
      
   2. What might be done to help her understand?
      
   3. Pat says that one third of a pizza can be more than half of a pizza, and he draws the following pictures to prove it. What are some of the issues Pat has raised?
3. You want to introduce fractions to second grade students. What are some tasks you might design in order to find out what knowledge of fractions they already possess?
   
4. For each of the following, draw a picture of a model you might use to help your students represent the expression:
   1. 5/6 - ½
      
   2. 1/5 + ½
      
   3. 2/5 x 5/6
      
   4. 1 3/4 ÷ 1/8
5. Write a word problem for each of the four expressions in #5.
   
6. Jeff has half a pizza left over in the refrigerator. For lunch, he eats one third of his leftover pizza. How much pizza does Jeff have left?
   
7. Three fourths of the seventh grade class went to the football game. Of the ones who went to the game, one third went by car. What part of all of the seventh graders went to the game by car? Explain your answer.
   
8. Write out your thinking process as you work through the following problem. If you get stuck as you are thinking about the problem, write "stuck." If you attempt a solution and it leads you to a dead end, do not erase what you did. Instead, explain how you knew the path lead you to a dead end. When you arrive at a solution, convince yourself that the solution makes sense. If you arrive at a solution that you are not sure is correct, explain your doubts.
   
9. Building upon the reasoning you used in #9, explain why, when we divide fractions, we invert and multiply. Try to provide a conceptually-oriented explanation.

Read the following paper and answer the questions below.

Thompson, A. G., Philipp, R. A., Thompson, P. W., & Boyd, B. A. (1994). Calculational and conceptual orientations in teaching mathematics. In D. B. Aichele & A. F. Coxford (Eds.), Professional development for teachers of mathematics, (pp. 79-92). Reston, VA: NCTM.

1. Three orientations were either discussed or referred to in the paper: Conceptual, calculational, and computational (Note: Computational was only referred to in a footnote.) Briefly describe the conceptual and calculational orientations, and explain how they differ from each other.
   
2. Two orientations were discussed in the paper, conceptual and calculational. Which of these best characterizes your orientation? Which of these best characterizes your teachers orientations? What orientation do you hope to have as a teacher?
   
3. Consider the following question:
   Approximately how many times has your heart beat since you were born?
   1. Provide a calculationally oriented explanation for your solution to this question.
      
   2. Provide a conceptually oriented explanation for your solution to this question.
      • Consider the following problem:

        Michael goes to the store to purchase some sliced turkey. He would like to buy 1/4 pound of turkey, because that is how much his diet allows him to eat for lunch. The woman behind the counter asks him whether 1/3 of a pound would be acceptable, since the turkey is sliced into equal sized round pieces so that 3 slices equals 1/3 of a pound. Michael buys the 3 slices of turkey and goes home to have lunch. The diet Michael is following allows for 1/4 pound of turkey for lunch, and it is very important to Michael to follow his diet. How might he slice his turkey?

        
        
      • Provide a conceptually oriented explanation for your solution to this problem.

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