Go to https://sites.google.com/site/math105site/ for all of the modules I eventually wrote for Math 105.
Sabbatical Leave Report
Julie Harland
October 1996
I. Signature Page
See attached.
II. Restatement of Sabbatical Purpose
My purpose was to develop material that would facilitate collaborative learning in my Math 105 class (Math for Elementary School Teachers).
III. Restatement of Objectives
My main objective was to develop and write three complete, comprehensive chapters of original material for Math 105. These chapters would include discovery exercises (designed for students to do while working in collaborative learning groups), thorough explanations of the mathematics involved and complete solutions. In order to present the material in a professional manner, I also planned to study word processing and graphics software as well as learn how to integrate word processing and graphics in my work.
IV. Description of Activities
Because I was on maternity leave the year prior to my sabbatical leave start date, I was able to do some research and take the time to study some word processing and graphics ahead of time. Specifically, I browsed and studied several books that are being used to teach the equivalent of Math 105 at other colleges. I also worked on word processing and graphics tutorials. I started jotting down ideas for group activities and tried using graphics to make "manipulatives". Manipulatives are objects the students use in order to explore and discover mathematical concepts. This gives students a hands-on experience, which in turn brings the concepts to life and enhances students' understanding. By August 1995, I started writing my first chapter. Originally, my intention was to write chapters that were similar in length to those found in commonly used texts. However, as I began my research and writing, I realized that the needs of Math 105, which employ collaborative learning exclusively, necessitated a more module-oriented approach to writing—modules containing exposition mixed with exercises, material cards, workbook segments, review problems and complete solutions. I worked to develop modules that are self-contained, detailed and comprehensive. To achieve the quality and depth that I wanted to produce, I focused my sabbatical time purely on writing. (In my original sabbatical leave application, I had intended to devote 25% of my sabbatical leave time to observing collaborative learning classrooms. In February 1996, I requested a change to focus all of my time on writing, which was approved by the president, Tim Dong. See the attached letter.)
In writing my first module, I employed techniques learned from the computer tutorials and as a result, I was able to format the modules in a professional manner. By the second module, I had decided to make Material Cards a separate entity. Material Cards are templates for manipulatives the students make and use in order to work through the various exercises. For maximum effectiveness, each individual student will have their own set of manipulatives for out of class use and possibly for future use as an educator since manipulatives are now being used extensively in K-6.
I was able to finish three complete modules by August 1996. Each module is comprehensive and self-contained, comprised of a table of contents, four or five exercise sets, review exercises and complete solutions. These cover more material than the three chapters I had intended to replace out of the book we have been using. There are over 200 pages of writing and graphics. I also created 35 Material Cards. This is a separate packet of material to be used with the modules. Needless to say, I expended more than a semester's worth of time developing these materials for Math 105.
I am attaching the table of contents and a sample page from each module to this report. In addition, I would be happy to demonstrate some uses of the manipulatives made from the Material Cards to anyone interested.
V. Contribution to District
Writing these modules has contributed greatly to my own professional and personal growth. I am much more competent at using the computer to develop documents. This will enhance my work as an instructor as well as my role as a contributing member to the college as a whole.
The most important investment any instructor can make is to improve the quality of learning for his or her students. I have achieved this goal through my sabbatical. Although developing these detailed modules required significantly more time than writing traditional expository chapters, I believe the modules are of greater value to Math 105 students and instructors than the textbooks commonly used in this course. My students are presently reaping the benefits of my work. The bookstore printed the materials I developed and sold them to students at the beginning of this semester. Thus, students are using new, innovative, up-to-date materials in class right now. This is the smoothest semester I have ever experienced. As usual, the students are enjoying working together in collaborative learning groups. The difference is that they feel they really understand the material. Although the exercises are in many cases more challenging than those found in traditional texts, there are fewer questions in class because of the thoroughness of the modules. It is important to note that my students really experience the connection between their academic work here and their future careers since most elementary school teachers now use collaborative exercises and manipulatives in the classroom. In summary, I am very satisfied with the results I am seeing in my students. I only wish that I had had the time to complete the entire set of modules necessary so that no other book would be needed for Math 105 and for its successor course, Math 106. Creating more modules is a long term goal of mine and I plan to continue writing whenever I have the time.
The individual modules I wrote may eventually be used in the Math Learning Center for students needing remedial work in certain areas, or for Math 105 students who wish to take this course self-paced or in the summer. In this capacity, the writing done on this sabbatical may end up serving an even broader population of students than originally intended.
Both the full-time and part-time colleagues in my department may use the materials I developed during this sabbatical when they teach Math 105. In addition, I will share my writing experiences with anyone interested.
VI. Additional Sabbatical Leave Activities
My eldest son, Jakob, was in first grade while I worked on this sabbatical. His school was piloting a completely new math curriculum which makes extensive use of manipulatives. I volunteered in his classroom during his mathematics lesson for about an hour and a half once a week. During this time, the teacher generally turned the class over to me. This was an excellent opportunity for me to experience and observe the use of manipulatives in an elementary school setting. I also served on a parent/teacher school site committee designed to study the program. Volunteering in the classroom has given me valuable insight and information to bring back to my students who are working toward becoming elementary school teachers.
VII. Documentation of Leave
I am submitting one copy of the three modules that I created, developed and wrote — Set Theory, Counting and Numerals and Addition and Subtraction. I am also submitting one package of Material Cards.
My educational and professional growth has been greatly enhanced in many ways as a result of this sabbatical. My skills on the computer have improved significantly. I understand what it takes to write modules for math. I am rejuvenated after taking a break from classroom teaching which I feel improves my performance as a teacher. As for my Math 105 students, I am very fortunate that they like and appreciate the materials I have written. At first, I was slightly apprehensive about using my modules, but the students are asking me to hurry and write some more!
All in all, the taking of this sabbatical has been a wonderful experience and I appreciate that I was given this opportunity. I am grateful to all of the people at MiraCosta who supported me and helped to make this sabbatical possible.
Date: February 13, 1996
To: PAC
From: Julie Harland
Re: Sabbatical
I am requesting a minor change in my sabbatical.
My approved sabbatical plan for this year stipulates that I spend 75% of my time writing chapters for a Math 105/106 text and the other 25% of my time observing collaborative learning classrooms. Now that I have written a fair amount, I believe it will be of greatest benefit to my students if I spend all of my time on writing.
Originally, my intention was to write chapters that were similar in length to those found in commonly used texts. However, as I began my research and writing, I realized that the needs of Math 105/106, which employ collaborative learning exclusively, necessitate a more module-oriented approach to writing—modules containing exposition mixed with exercises, material cards, workbook segments, review problems and complete solutions. I am submitting a sample of my work—a hands-on workbook on Set Theory (I am still doing some minor editing on this module). I will be using this and other modules I develop when I return to teaching next year.
Because the modules I have been developing are self-contained, in-depth and comprehensive, I believe they will be of greater value to Math 105/106 students and instructors than traditional expository chapters found in most texts. In addition, these individual modules may eventually be used in the Math Learning Center for students needing remedial work in certain areas, or for Math 105/106 students who wish to take this course self-paced or in the summer.
I still feel that visiting other collaborative learning environments would be an interesting and beneficial experience for me. However, I have found that developing detailed modules requires significantly more time than writing traditional expository chapters. At this point, I believe that the best way to meet the needs of my students is to focus the remainder of my sabbatical time purely on writing. In doing so, I could maintain the same quality and depth that I have been able to produce so far.
I thank you for your time in considering this request and await your response.
Table of Contents
and
Sample Pages
from modules on
Set Theory
Counting and Numerals
and
Addition and Subtraction
as part of the Sabbatical Report by Julie Harland
Set Theory
Table of Contents
Exercise Set 1: The Basics of Sets................................ 1
Exercise Set 2: Venn Diagrams.................................. 15
Exercise Set 3: More About Sets................................ 27
Exercise Set 4: Survey Problems................................. 43
Important Terms and Concepts.................................... 55
Review Exercises......................................................... 56
Venn Diagram Templates............................................ 65
Selected Solutions to Exercises.................................... 69
Below is a list of Material Cards to be used with this workbook. You will be using these to make your own manipulatives to do hands-on activities in some of the exercise sets. You should prepare them ahead of time. At the beginning of each exercise set, you are told which material cards you will be using for that set. You may also need scissors, tape and/or colored pencils, pens, markers or crayons.
Material Card 1.................................... Coins
Material Card 2.................................... Red A-blocks
Material Card 3.................................... Yellow A-blocks
Material Card 4.................................... Green A-blocks
Material Card 5................................... Blue A-blocks
Material Card 6................................... Blank A-blocks
Material Card 7................................... Value Label Cards
6. Name some combinations that leave exactly one piece outside the three sets.
[Hint 1: Pick a piece you want to be left out. Hint 2: Use some NOT labels.]
7. Write down some combinations that leave the most number of pieces outside the sets. What do you think is the most number of pieces that can be left outside the three circles?
8. Take one of your blank value label cards and on it, write
(for the UNION of BLUE and SQUARE). Take the label cards NOT BLUE () and NOT SQUARE (
) from your set. On the poster, put each label on one of the sets. Place each A-block in the correct region.
a. On the Venn diagram below, show where you placed each A-Block.
b. Look at the set BLUE UNION SQUARE (
). Look at all of the pieces outside the set so you can list the elements in :
__________________________________________________________________
c. Look at the intersection of NOT BLUE and NOT SQUARE , and list its
elements : ___________________________________________________________
d. What do you notice about the elements listed in part a and part b above?
e. Can you write an equation from what you noticed? ________________________
Counting and Numerals
Table of Contents
Exercise Set 1: Numbers and Numerals....................... 1
Exercise Set 2: Numeration Systems.......................... 13
Exercise Set 3: Trading and Place Value................... 21
Exercise Set 4: Other Base Systems........................... 31
Important Terms and Concepts.................................... 43
Review Exercises......................................................... 44
Selected Solutions to Exercises.................................... 51
Below is a list of Material Cards to be used with this workbook. You will be using these to make your own manipulatives to do hands-on activities in some of the exercise sets. You should prepare them ahead of time. At the beginning of each exercise set, you are told which material cards you will be using for that set. You may also need scissors, tape and/or colored pencils, pens, markers or crayons.
Material Cards 2-5................................ A-blocks
Material Card 8.................................... Models for Base Two
Material Card 9.................................... Unit Blocks
Material Cards 10-11............................ Base Two Blocks
Material Card 12.................................. Base Three Blocks
Material Card 13.................................. Base Four Blocks
So you won't have to keep turning back the page to remember these symbols, here again are the Chinese numerals you probably need to look at to do the following exercise.
2.__ Rewrite each Chinese numeral in Hindu-Arabic.
a.________ b.________ c.________ d._______ e._______
Notice in the Chinese system that numbers over ten have symbols written in pairs. To write 800, you must write the symbol for 8 and 100. Although the multiplicative principle allows you to write down less symbols than a simple additive system (for most numerals, at least), a further simplification would allow us to skip writing the second numeral of each pair. This would work if we used the position of the symbol to indicate the size of that group (10, 100, 1000, etc.) This type of system is called a positional numeration system. In order to keep track of a position where no digit is used, a symbol for zero is necessary. Although the Chinese system doesn't need a symbol for zero, a circle was introduced to represent zero in the 1200's.
The final system we'll learn about in this Exercise Set uses a positional system and is similar to the Chinese system in that the symbols for the numerals are written from top to bottom. Mayan Numerals were developed by the Mayan priests of southern Mexico and Central America around 300B.C. It is believed to be the earliest positional numeration system incorporating a zero and using it for a placeholder.
3.__ Some Mayan Numerals are shown below. Try to figure out the pattern and then fill in the missing numerals.
Addition and Subtraction
Table of Contents
Exercise Set 1: Definition and Properties.................... 1
Exercise Set 2: Combining........................................... 11
Exercise Set 3: Addition Algorithms........................... 23
Exercise Set 4: Subtraction......................................... 37
Exercise Set 5: Subtraction Algorithms...................... 53
Important Terms and Concepts.................................... 64
Review Exercises......................................................... 65
Selected Solutions to Exercises.................................... 76
Below is a list of Material Cards to be used with this workbook. You will be using these to make your own manipulatives to do hands-on activities in some of the exercise sets. You should prepare them ahead of time. At the beginning of each exercise set, you are told which material cards you will be using for that set. You may also need scissors, tape and/or colored pencils, pens, markers or crayons.
Material Cards 1................................... Coins
Material Cards 2 - 5.............................. A-Blocks
Material Card 9 - 22............................. Base Blocks
Material Cards 23 - 34.......................... Centimeter Strips
Material Card 35.................................. Base Ten Money
6.__ Take out a set of Base Four Blocks. Write 15 in Base Four: ____________________
___ Make a pile, called Pile A, using the Base Four Blocks to represent 15. Now, write the Base Four numeral for 13: Using the Base Four Blocks, make a pile, called Pile B, to represent 13. Combine the blocks from Pile A and Pile B to form one big pile of blocks. You are forming the union of Pile A and Pile B! Make all possible exchanges and then write the Base Four numeral that represents the number of units in the combined pile here: ____________
Combining the two piles is the same as forming the sum of the two numerals. You have just added two numbers together in a different base.
Write the addition problem you have just performed in Base Four below.
________________________ + _____________________ = _____________________
Now, convert the sum to Base Ten: _____ It should be 28 since 15 + 13 = 28. Is it?
7.__ Take out a set of Base Seven Blocks. Write 15 in Base Seven : __________________
___ Make a pile, called Pile A, using the Base Seven Blocks to represent 15. Now, write the Base Seven numeral for 13: Using the Base Seven Blocks, make a pile, called Pile B, to represent 13. Combine the blocks from Pile A and Pile B to form one big pile of blocks. You are forming the union of Pile A and Pile B! Make all possible exchanges and then write the Base Seven numeral that represents the number of units in the combined pile here: ____________
Write the addition problem you have just performed in Base Seven below.
________________________ + _____________________ = _____________________
Now, convert the sum to Base Ten: _____ It should be 28 since 15 + 13 = 28. Is it?
8.__ Take out a set of Base Two Blocks. Write 15 in Base Two : ____________________
___ Using the Base Two Blocks, make a pile, called Pile B, to represent 15. Now, write the Base Two numeral for 13: Using the Base Two Blocks, make a pile, called Pile B, to represent 13. Combine the blocks from Pile A and Pile B to form one big pile of blocks. You are forming the union of Pile A and Pile B! Make all possible exchanges and then write the Base Two numeral that represents the number of units in the combined pile here: ____________
Write the addition problem you have just performed in Base Two below.
________________________ + _____________________ = _____________________
Now, convert the sum to Base Ten: _____ It should be 28 since 15 + 13 = 28. Is it?