Math 100B
MATHEMATICS 100B: Winter 2004
MATHEMATICS FOR ELEMENTARY TEACHING
Instructor: Brie Finegold
Class Meetings: Tuesday and Thursday 5:30 – 6:45 in Trailer 940
Office: 6432H South Hall Graduate Tower
Office Hours: Monday 10:15-11:15 AM
Tuesday 3:30-4:30
Wednesday 7:50-8:50AM
Final Exam: June 5th
Texts: (first three are required)
Math 100AB Reader, Available at Alternative Copy Shop
Bassarear, T. (2001). Mathematics for Elementary School Teachers. Boston: Houghton Mifflin
Fosnot, C. Dolk, M. , Young Mathematicians at Work : Constructing Multiplication and Division (Note: Not to be confused with the volume on addition and subtraction.)
Available on the Web at: National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics. Reston, VA: NCTM
Overview of the Course: The focus of Math 100B will be on integers, rational numbers, proportional reasoning, and an integrated treatment of algebra and geometry.
Much of the time in class meetings will consist of group explorations. Additionally, students are expected complete all assignments, most of which will require considerable out-of-class work. Assignments will include reading assignments and solving problems from the basic text, some observations from CD-Rom clips, and short reflective papers. Assessment will be based on both in-class and out-of-class assignments.
Content Coverage/Exam Schedule: See attached Tentative Class Activities and Assignment Schedule
Grading: The course is designed for students who are contemplating becoming an elementary school teacher. The fact that it is a pass/no-pass course has nothing to do with level of work expected of the student. A few years ago, a number of students with strong backgrounds in mathematics took this course to raise their GPA--even though they would not receive credit for satisfying major requirements. As a result, the classes became crowded with students who had no intention of becoming teachers. Students who really needed the course frequently were unable to take the course because of the overcrowding.
Your final grade in the course will be based on your performance on homework problems, reflective writings, group participation and presentations, a portfolio, quizzes, and a final examination.
The course grade will be based on the following:
Homework20 %
Group participation, presentations
and write-ups 20 %
Reflective Writing20 %
Quizzes (2 of 3 quizzes, 10% each)20 %
Portfolio and Self-Assessment 20 %
100 %
A Five-Point Grading Rubric will be applied to some exam problems and some of the problems occurring on homework and projects as follows:
5Complete and correct response including extensions and connections
4 Complete and correct response
3Substantially complete and correct response
2 Some partial success
1 Engaged task with little success
0 No response. Did not engage question
A passing score consists of receiving a majority of 3 or better on an assignment’s tasks.
Attendance:Attendance is required. Group activity will be a regular part of this class. There are unforeseen emergencies that do come up. However, anyone missing the class more than 4 times during the term will not receive credit for the course. Group activity will usually be collected and will determine 20% of the student's course grade. There is no make-up of missed group activity.
Student Preparation: You will be expected to read and keep current with assignments given in the course. Homework assignments will be collected twice weekly and a reflective writing will be due each Friday. Occasionally group presentations will also be required. Keeping current with assigned work and presentations will be important in this class, and no late homework will be accepted unless prior arrangements are made with the instructor.
Math Lab: South Hall 1607, open Monday through Friday, 12:00 - 5:00 p.m. A Teaching Assistant will be available to assist you with mathematical questions you may have.
Office Hours: Please take advantage of my office hours. I actually look forward to students dropping by and discussing mathematics. Don't be shy in making regular appearances.
Pedagogy: My role will be that of a facilitator to our construction of learning during the next ten weeks. This is not a methods course. Again, I repeat, this is not a methods course. It is a mathematics content course. However, you will find that the way this class is taught will be much different than the typical lecture format of many mathematics classes--though there will be some lectures and videos interspersed during the next ten weeks. This is an important shift in teaching practice for many of us. Often, portions of our class time will be spent working in groups in an effort to develop solution strategies for various problems and explorations. Courses in pedagogy (teaching methods) will occur during your coursework within the credential program.
Overall Philosophy: I require written explanations of the mathematics you are studying on all homework, group investigations, quizzes, and exams. This course emphasizes the conceptual framework of mathematics and is designed to avoid the "turn the crank" style of computation that is typical of many mathematics courses. Solution of homework problems will require careful thinking about what is really going on. You will not always be able to solve the problems by imitating a procedure found in class or in the book. Warning! The answers you may find in "solutions or answer books" often represent only partial solutions. These are usually not sufficient to receive complete credit. (Refer to the grading rubric above.) Also: Just getting an answer is not enough, you are expected to explain connections between ideas and think about extensions of you work. You may wish to use answers as hints for certain problems, but they are not models for complete solutions. Again I emphasize, if you are stuck, work with a classmate, come to the Math Lab, see one of the assistants, send me e-mail, and bring questions to class meetings. On all assignments and exams, it is crucial that you explain, in complete sentences, what you are thinking. It is possible to receive a poor score for a correct answer if you do not communicate to me your ideas. On the other hand, a clear exposition with a minor computational error can receive a good score. You are expected to read the text carefully. Not every detail will be covered in class meetings (in fact quite a few may not), and some different issues and examples raised in the text could show up on the exams or on assignments.
Video Work: During Math 100B, several class meeting will be devoted to watching and discussing video clips of elementary students doing mathematics. This is not intended to prescribe how to teach (as I said this is not a methods course). Instead, we want you to be aware of how students can develop their mathematical thinking by using inquiry in instruction. Also, we want you to think about the role of context in posing mathematical questions, and to think about how instructional decisions are made and can be based upon student responses. (Instructional decisions that are made by turning to the next page in the book typically do not meet students’ needs.) You should try to think about relationships between what you see in the videos and your study of mathematics in this course. Reflective writings will expect you to consider these issues.
A Final Comment:I want this to be a successful and enjoyable learning experience for you. During the next ten weeks, I hope that you will reflect carefully on any plans you might have for entering teaching at the elementary level. The present climate of reform in mathematics education offers many opportunities and challenges for teachers. On occasions we will discuss these issues. The education of our youth is a tremendously complex process, and I will do everything I can to help you get started in this profession. Please feel free to contact me if you have questions or want to share insights. Using e-mail is probably the most efficient way of reaching me. It's also a way of getting clarification on homework problems. Finally, the mathematics department offers paid summer teaching internships for UCSB undergraduates. Placements in elementary classrooms are available and you do not have to be a math major (in fact we want students from all majors to be successful in elementary mathematics teaching!) Many undergraduates have found this enormously valuable preparation for a credential program. Talk to me about it any time.
Math 100B
Mathematics for Elementary Teaching
The Number System, Proportional Reasoning, Geometry
Proportional Reasoning
3/30Course expectations: Exploration: Factors, GCF and LCM
Distribute Cuisenaire rods for homework problems.
4/1Finish Factor Work Problem Set 1 Due
4/6 Partative/Quotative Division
4/8Video Clip#1 Soda machine Problem Set 2 Due
4/13Exploration: Partitioning WholesQUIZ 1
4/15Partitioning Wholes Continued Problem Set 3 Due
4/20Video Clip#2 Making Sense of Fractions First Reflective Writing Due
4/22Operations with Fractions Problem Set 4 Due
4/27Operations with Fractions
4/29Explorations: Proportional Reasoning and Functions, Reducing, Enlarging and Percents Problem Set 5 Due
5/4Video Clip #3: The Ratio Table
5/6Ratio Tables for Fractions/Decimals/percents
Second Reflective Writing Due, Problem Set 6 Due
5/11Exploration: Growth Patterns and Snakewood Patterns
5/13Comparing Quantities Problem Set 7 Due
5/18Expl. 8.1: Geoboard Explorations (Not in Reader)
5/20More Geoboard Explorations, Problem Set 8 Due
5/25 Tesselations Third Reflective Writing Due
5/273D to 2D Explorations, QUIZ 3 Problem Set 9 Due
6/13D to 2D Explorations
6/3Closing discussion Portfolio, Self-AssessmentDue
NOTE: SUPPLEMENTAL PROBLEMS MAY BE ASSIGNED. DON'T FORGET TO HAND THESE IN WITH YOUR HOMEWORK ASSIGNMENT!
A note on evaluating homework assignments: Typically, four problems from the problems will be selected for grading from the exercises. Each problem will be scored using the four-point rubric mentioned in the syllabus. Also, when appropriate, please assemble each problem set into two parts—each with your name on them—since they will be scored by different readers.
Problem Set 1: Be sure to Pick Up Sample Cuisenaire rods in class!
Part I. Read Chapter 3 from Fosnot and Dolk. The first two chapters are based on the same template as are Chapters 1 and 2 from the first Addition and Subtraction book BUT, the examples are different. You may want to glance at them. Now reread pages 43-44 again, and create four different subdivisions of an open array to use as method to multiply 16 x 24.
Part II Read pp. 205-225 from Bassarear. Problem solutions to be turned in:
Exercises pp. 212-214: 5, 7, 10, 12, 14, 19.
Exercises pp. 225-226: 5, 7, 11, 17-19.
Problem Set 2
Read Chapters 4 and 5 from Fosnot and Dolk-Multiplication.
(a) View the following Clip from the Soda Machine CD page 22 (This is the Sprite Machine and is clip #23 if viewed separately). How does this new problem facilitate development of further strategies for division?
(b) Write a short paragraph, describing their strategies and growth over time, for at least three of the pairs of students as they work on all three soda machine problems: Indya and Ruby (#4,5,6,24,25), Johana and Michael (#15, 26), Ashley and Lucas (#8, 9, 27), Carlos and Elvin (# 14, 29)
Problem Set 3
Read pp. 239-251 from Bassarear. Problem solutions to be turned in:
Exercises 252-254: 2-5, 8, 24, 30 (30d is a bonus problem)
Problem Set 4
Part I. Read the first section from the Dolk Fractions book that included in your reader. This includes a written description of Carol Moseson’s class where they work on the submarine sandwich problem.
(a) How do you think these students would represent subdividing 5 submarine sandwiches between 8 kids? How about subdividing 7 submarine sandwiches between 10 kids?
(b) Use your representations in (a) to determine which share is larger (Do not compare them using decimals or other tools you knew about before learning about Carol’s class.)
(c) On p 11, Fig. 1.3 shows John and Jennifer’s paper. They have found the shortcut for finding the amounts each group gets, and they have indicated they know group 3 gets the most. But they don’t indicate how they know this latter fact. How do you suppose they might, and how could their representation be used to explain this?
Part II. Read pp. 255-273 from Bassarear.Problem solutions to be turned in:
Exercises pp 274-276: Identify the representation suggested by the context of each problem and solve the problem: 5, 6, 7, 10, 12, 13, 15, 29, 30 Now solve: 32, 33
Problem Set 5
Read and study Section 276-299. Problem solutions to be turned in:
Exercises 299-301: 2a, c, 4a, e, 8, 9, 15, 16, 18, 34, 38
Problem Set 6
Part I. Read the first section from the Dolk Fractions book that included in your reader. It includes the ratio table presentation from Joel’s class. Write a paragraph explaining how ratio tables could have been used to solve Investigation 6.7 on page 316-317 in Bassarear’s book. How does this approach compare to the two approaches given in Bassarear’s book?
Part II. Read and study Section 305-320. Problem solutions to be turned in:
Exercises 320-322: 9, 13, 14, 20, 24, 34
Problem Set 7
Read and study Section 322-338. Problem solutions to be turned in:
Exercises 338-341: 1a, g, 3, 5, 39, 43
Problem Set 8
Part I. Read Chapters 6 and 7 from the Fosnot and Dolk Multiplication Book. Also look over the landscape of learning on pages 136-137. Identify at least two big ideas, two strategies and two representations in these sections that are critical to making sense about fractions and proportional reasoning. (The point here is that a robust understanding of multiplication and division and its representations are crucial to much of the work done in Math 100B—which explains why we continued to use the multiplication and division book during this class even though we moved on to further material!)
Part II. Read Bassarear 457-473. Problem solutions to be turned in:
Exercises 474-476: 4, 9, 14, 20.
Problem Set 9
Read Bassarear 474-485. Problem solutions to be turned in:
Exercises 485-487: 2, 14, 10, 15, 17, 21
Math 100 B First Reflective Writing: Supporting Number Understanding with Geometric Representations
NOTE: You must visit the Curriculum Lab in the UCSB Library to complete this assignment. So do not put it off until the last minute.
First you should review the Soda Machine Context and the student work on your CD rom.
Choose two student pairs on the CDwhose work was quite different and explain how geometric representations supported their development of understanding partative and quotative division. You must explain how the geometry and number understanding relate to each other.
Second, you should go to the Curriculum Laboratory on the first floor of the UCSB Library Find the shelves containing the current California mathematics adoption. (In the back on your left in the second or third rack as you walk toward the window.) Find a fourth grade student text and look at the tasks they are assigned when introducing multiplication and/or division. Spend 5 minutes and jot down a few notes about what you find, and be sure write down the publisher and page numbers too for later reference. (Note: you cannot check these texts out—please return them to the shelf so your classmates can find them to.) Answer the following questions.
(i) Are students expected to develop their own representations that become tools for calculation?
(ii) In general, do most assignments expect student to “mathematize”? Or is the objective of the text to teach algorithms? Give your reasons.
Math 100B Second Reflective Writing: Models to Tools
One of the themes of the Fosnot-Dolk book (and videos) is that a good contextual task leads students to develop a model or a representation that is both a tool for solving the problem and enables them to construct deeper mathematical understandings. Then, early in Chapter 5, Fosnot and Dolk quote Gravemeijer:
“The shift from model of to model for concurs with a shift in the students thinking, from thinking about the modeled context situation, to a focus on mathematical relations.”
Fosnot and Dolk say that this shift is a “major landmark in mathematical development”. They discuss what they mean in more detail pp. 86-88 of the Multiplication and Division book. In class, and in your reading, we have developed several models for multiplication and division. Citing examples from the book or if you prefer, citing your experiences in this course, complete all three of the following:
(1) The congress following Joel’s class’ work on cat food prices is discussed on pages 45-50 of the Fraction book. Fosnot and Dolk say,
The purpose of the congress is not to get all children to the same point at the end but to explore connections among the solutions—to challenge each child yet keep the community as a whole moving toward the horizon.
Do you think Joel’s math congress accomplished their stated goal? Do the children move from a model of to a model for in their work with the ratio table?
(2) Second describe your favorite representation for multiplication or division. In a few sentences, say what it is and how it works. Give a context/problem that might lead students to use this representation to solve the problem. (Note: You do not give students the representation.) Then give an example of how this model can subsequently be used as a model for thinking about a particular mathematical relationship (so now, the question or relationship being considered is no longer in context.) For example, you could consider one or two of the strategies listed on p. 85 or one or two of the strategies or big ideas on the Landscape on pp 136-137.
Math 100B Final Reflective Writing
Read Chapter 8 on Assessment from Fosnot-Dolk.. On page 127 they say:
Assessment outcomes today not only define what will be taught but also are used as gates to educational programs and schools. They are used to determine how much federal money schools will get. They put teachers on the line for job security and promotion. They are even used to evaluate schools and districts, thus affecting property values and the demography of neighborhoods. They are a high-stakes game. And make no mistake, they drive instruction.
In California, the Academic Performance Index (API) is a number computed by the State Department of Education that ranks schools. It does take into account socio-economic factors, but until now, the sole measure of student achievement is the Stanford 9, a commercially produced multiple choice achievement test. (Also, the Standford 9 is a "norm referenced test, which means that instead of testing kids on topics from their instructional materials, the questions are spread out to ensure that a bell shaped curve of scores is attained and students can be ranked according to "percentiles".) The stakes are high: low performing schools can be taken over by the state, teachers whose students score high win extra $, and principals whose school's scores don't go up can be fired. By the way, California is not alone, with the President's new education plan, all kids in the country in grades 3-8 must take such tests.