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Augusta State University Department of Mathematics and Computer Science MATH 6241 * Understanding Numbers and Operations Fall 2009 *Allgood Hall E260

Instructor Linda Crawford, Ph.D. Allgood Hall N322 (706) 667-4477 lcrawfor@aug.edu (observe: it is lcrawfor-----there is no “d”!!)

Office Hours · Monday & Wednesday, 2:30-3:45 · Tuesday & Thursday, 11:30-12:30 · Other times by appointment

Course Description

This course focuses on an understanding of the major concepts of number and operations. Collaboration, critical thinking, hands-on manipulatives, and problem-based inquiry as methods of instruction and assessment will be used to examine what it means to teach and learn number concepts at the P-5 level.

Prerequisite: Permission of Instructor

Purpose of the Course

This course focuses on preparing P-5 mathematics specialist candidates to:

· Understand and use the major concepts of number and operations in mathematics for grades P-5.

· Solve problems using multiple strategies, manipulatives, and technological tools; interpret solutions; and determine reasonableness of answers and efficiency of methods.

· Nurture collaboration, critical thinking, hands-on exploration, manipulative use, problem-based inquiry, technology utilization, and activity implementation addressing various learning styles and multiple intelligences.

· Select and use a variety of formative and summative assessment techniques to monitor student progress, gauge students' mathematical understanding, and interpret school-based progress.

Course Objectives

The mathematics content covers three broad areas: pre-number/early number concepts; number basics which includes an overview of the relationships of the real numbers; and the four operations of addition, subtraction, multiplication, and division.

In the area of pre-number/ early number concepts, all students will be able to:

• Have knowledge of appropriate activities to develop young children’s skills in classification/sorting, patterns and comparisons.
• Appreciate conservation of number as a prerequisite to rational counting and know how to evaluate for conservation of number.
• Recognize number as an attribute of a set.
• Appreciate the value in group recognition of number as a learning tool.
• Differentiate between rote and rational counting.
• Use various counting strategies.
• Understand various interpretations of equivalence.
• Know the characteristics of our number system:
• understand the historical development (including the contribution of various cultures) and use of multiple representations of number and numeration systems
• compare and contrast the symbols used in each numeration system and understand why these particular symbols were used

• Use the language of mathematics to formulate accurate, precise, and pedagogically appropriate definitions of terms related to number and operations.

In the area of number basics all students will be able to:

• Classify (sort) numbers by their membership in subsets of the real numbers.
• Recognize and represent equivalent forms of a number (fractions, decimals and percents).
• Use various representations to model numbers.
• Compare and order a set of numbers from various subsets of the real numbers.

· Understand meanings and representations of numbers.

In the area of the four operations of addition, subtraction, multiplication and division all students will be able to:

• Develop a deep understanding of the meaning and representations for the four operations.
• Use and apply traditional algorithms and be able to explain why they make sense.
• Model, explain and develop a variety of alternative algorithms.
• Understand and apply properties of the number systems.
• Understand and use multiple strategies for estimation.
• Understand and use a variety of strategies for mental math:

o Justify mental computation techniques by employing the properties of arithmetic.

o Use mental imagery of number representations to compute.

o Use the properties of arithmetic to aid and structure the memorization of basic facts.

o Show the algebra underlying mental techniques by writing equations that go along with a mental method of computation.

• Analyze correct and incorrect solution strategies and explain why the method is correct or incorrect.
• Understand and apply inverse operations.
• Identify, explain and model multiple strategies and approaches in problem solving.
• Use the four operations with whole numbers, fractions and decimals in applied situations involving measurement of various attributes (operations should involve distance, area, volume, capacity, weight, time, money, etc.).
• Engage in problem solving activities that develop the skills of communicating, justifying, reasoning and clarifying, i.e., learning how to tell “why” they solved a problem the way they did.

In addition, students shall have experience:

• Reasoning from a definition.
• Comparing two or more definitions and showing that they are (or are not) equivalent, e.g., splitting fairly into two groups vs. into groups of two.
• Making a definition more precise, e.g., elaborating what is meant by "fairly" in the previous definitions.
• Expand the domain of a definition, e.g., the definition of "even" from natural numbers to integers
• Choosing a definition that is pedagogically appropriate for a particular group of students
• Recognizing the need for undefined terms.

The pedagogical and professional development objectives for the course include acquiring the ability to:

• Demonstrate knowledge of appropriate concrete materials and technological tools for learning mathematics.
• Determine the suitability of the wide variety of available mathematics curricula and teaching materials.
• Demonstrate knowledge of different types of mathematical instructional strategies.

· Demonstrate knowledge of student diversity to affirm and support full participation and continued study of mathematics by all students.

Learning Outcomes for the Course

All students will learn to:

· Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

· Understand meanings of operations and how they relate to one another.

· Compute fluently and make reasonable estimates.

· Demonstrate a deep understanding of how P-5 students learn mathematics and of the pedagogical content knowledge appropriate to P-5 mathematics teaching.

Supplies

· Required Textbook: Math Matters: Understanding the Math You Teach, Grades K–8, 2^nd edition, by Suzanne H. Chapin and Art Johnson (ISBN: 978-0-941355-71-1)

• Colored pencils
• 3-ring binder
• Scientific or graphing calculator (to be used only when permitted)
• Optional Textbook: Principles and Standards for School Mathematics by the National Council of Teachers of Mathematics (2000). A 120-day free access to complete Principles and Standards for School Mathematics can be obtained at http://standards.nctm.org

Some of the assignments will be in PDF form so you will need Adobe Reader software—this can be downloaded free from http://www.adobe.com/products/acrobat/readstep2.html

I often use Windows Journal to grade assignments you submit electronically. You may need the computer program Windows Journal Viewer to open the returned assignment. You can download it free by clicking on Windows Journal Viewer.

Some assignments will be submitted through GeorgiaVIEW Vista (WebCT Vista). You can also access Vista through the My Courses tab on Pipeline. If you need help with Vista, visit the online support center.

If you need help with technology, check with the Information Technology Student Help Desk--either in person or by phone. The number is 706-737-1676. Information about the Help Desk can be found at http://www.aug.edu/its/Welcome.html.

If you have a disability and wish to receive accommodations in class, please apply with the Office of Disability Services.

Assignments to be turned in are due at the beginning of the class period. Put the assignment on my desk when you arrive for class. Even if you are absent, your assignment is still due at the beginning of class. If you will not be in class on a day that an assignment is due, you may email it to me or place it in my mailbox before class starts or send it with another student. A late assignment will be accepted only in extreme and documented situations.

If You Have to Miss a Class…

MATH 6241 class sessions are interactive, providing many opportunities for you to express your own ideas and to listen to the ideas of your fellow classmates. Much of what you learn in the course takes place by participating, sharing, and interacting with others through small-group and whole-group discussions. This kind of learning cannot take place if you are absent so regular attendance and punctuality are required.

Frequently, ideas that we introduce in one session are expanded upon and developed more fully in later sessions. Thus, every class session is important. However, if you find that you are unable to attend a particular class session or might miss a part of a session (by coming late or leaving early), please contact me as soon as possible. Make arrangements to turn in assignments if you are going to be absent—even if you are absent you are expected to turn the work in when it is due. You should ask a classmate to obtain any handouts given out during the class you will miss—do not ask me for handouts that you miss. You are also responsible for any announcements made during the class. Remember, you are going to be a teacher—you must take care of your obligations!

Prior to returning after a missed class you are responsible for meeting with a classmate(s) to discuss the class session you missed. After this meeting you must write a 300-500 word “ Missed Class Paper” (typed) which includes the names of your classmates with whom you met, a description of the activities of the class you missed and how you engaged in these activities with your classmates, any research you might have done (use your book and the web as resources), an explanation of your understanding of the mathematical ideas investigated in class, any insights you gained, and any questions you have about these mathematical ideas. The paper is not to be merely a list of what was done in class but instead shows your effort at making sense of these ideas and what learning you have gained. This paper as well as any accompanying work from the missed class is to be handed in at the beginning of the next day’s class. Two points are deducted from your participation grade for each absence; one point will be returned if your “Missed Class Paper” meets the criteria outlined above.

Roll will be taken; any student who is absent more than 10% of the class time may be dropped with a WF. Excused absences count toward the 10%.

The percentages to determine your course grade:

· Midterm exam—Wed, Oct. 7	25%
· Other written or presented assignments—for ex, homework, quizzes, solutions of problems, reviews of websites or journal articles	15%
· Ongoing assignments—for ex, journal of open-ended reflections & projects	20%
· Participation—click here for more information	10%
· Comprehensive final exam--Wed, Dec 9, 6:00-8:00 PM	30%

Other Written or Presented Assignments

The written or presented assignments may include quizzes, homework, in-class presentations, out-of-of class projects, reflective writings, reactions to readings, analyses of student work, analyses of mathematics lessons, solutions of problems, etc.

Homework

You will be assigned reading, writing, questions, and problems to be completed for homework. All in-class activities will be based on the assumption that the required homework assignments and readings have been completed. This does not mean that all of your answers have to be perfectly correct. It does mean that you should have thought hard about each problem, made several attempts at solving it, and developed questions and conclusions about your solution strategies. Not every homework assignment will be collected and graded. Most homework assignments are listed on the assignment page; however, you are responsible for any assignment announced in class and not listed on this page.

Class Participation and In-Class Activities
Much of the success of this course depends on your level of interaction and participation throughout the semester. As you will soon discover, we will spend most of our class time sharing ideas, solution strategies, insights, and questions. During class sessions, I will assess both your preparation for class (e.g. whether you completed readings and assignments) and the quality of your participation in course activities by observing and interacting with you. I will be paying particular attention to your willingness to listen, to discuss, and to contribute to whole-class and group activities. Clearly, successful participation in this sort of class depends upon regular attendance.

Each class period will generally include a discussion of the homework and a review of the activities from the previous class. Although the discussion will take different forms on different occasions, it will always be the case that your ideas, strategies, and questions will guide the discussion. Sometimes, you will be asked to present a problem to the class. Other times, you may be asked to share your work in a small group. Other times, you and a small group of your classmates may work on a new problem related to your homework. While I promise to support you in finding answers to your questions, please be warned that my support will NOT consist of simply explaining solutions of problems to you. My job is to help you develop meaningful understandings for yourself, with the help of your classmates. Because this is a student-centered class, it is of utmost importance that you attempt all of the homework problems before class and do the assigned readings so that you can participate in the discussion. Satisfactory participation in this part of class means that you are willing to share your thought processes, questions, and solutions with the class (even when you don't think you have "the right answer") and that you also support your classmates as they participate.

During each class you will generally engage in some form of investigation of a mathematical topic. Typically, you will work cooperatively with 3 or 4 of your classmates, using various curriculum materials to guide your work. This in-class work will provide a conceptual basis for your understanding of the course material. Because your subsequent readings and homework assignments will build on these investigations, they require your careful attention. On occasion, without announcement, your in-class investigations may be collected. Also, you should be prepared to provide a written reflection on the in-class work. Some class time will be spent discussing questions and ideas that arise from the group investigations. It is imperative that you spend time outside of class reflecting on the group activities so that you fully understand the concepts. Merely hearing another’s explanation is no guarantee that you understand.

Your participation grade is determined using the Rubric for Participation. Two points will be subtracted for any absence (you may receive a refund of 1 point—see the section “If you Have to Miss a Class”).

Course Notebook and Reflections

You should organize all materials (handouts, class notes, homework, readings, writings, tests) in a 3-ring binder. This notebook will be a record of your work in the course and will also serve as a tool for reflection. It will also be a valuable resource to you when you begin teaching.

To assess your own growth as a mathematics educator you may be asked to reflect on your progress in developing a deeper understanding of number and operations. Because your notebook chronicles your progress in the course, it is your primary resource for writing your reflections. You should record those important moments when you experience a break-through in your understanding.

Projects

· Analysis of a Mathematics Video Lesson--To demonstrate your understanding of how to support students’ development of a mathematical concept in number and operations, you will watch a videotaped mathematics lesson and reflect on your observations.

· Analysis of a Number & Operations Instructional Unit—You will select and analyze a unit on Number & Operations from the Mathematics Frameworks site of the Georgia Performance Standards.

Professional Organizations

You are encouraged to join the following professional organizations:

• Georgia Council of Teachers of Mathematics (GCTM) at http://www.gctm.org (GCTM student membership is free if you’ve never taught before; otherwise, it is $20.)
• National Council of Teachers of Mathematics (NCTM) at http://www.nctm.org/membership/ ($39 student e-membership includes your choice of journal--Teaching Children Mathematics is recommended for K-5 teachers)--Go to sampler.nctm.org to register for 120 days of free access to the Prospective Teacher Sampler of resources ANDfor full access to the online version of Principles and Standards for School Mathematics.

Classroom Policies

· It is a policy of Allgood Hall that food and drink are not allowed in the classrooms so cups, bottles, etc. should be capped and stored in your bag. Furthermore, food and drink interfere with group activities and the use of manipulatives.

· It is distracting to me and disruptive to the class activities if you leave the classroom during the class period. Thus, I ask you to refrain from "taking a break" except for an emergency.

· Visitors, including children, are not permitted without my prior permission.

· You are expected to check your campus email regularly.

Academic honesty

Cheating will not be tolerated. This pertains not only to in-class work but to outside assignments as well-any assignment that you submit as your own should be a report of YOUR thinking. Any student who is caught cheating will face serious consequences. You should read ASU's statement on academic honesty in the catalog.

Dates to Remember

Mon-Tues, Sept 7-8 Wed-Fri, Nov 25-27	Holidays
Mon, Oct 12	Midterm date (if you plan to withdraw, do so no later than this date)
Thurs, Dec 3	Fall semester classes end
Wed, Dec 9, 6:00-8:00 PM	Final exam
October 14-16 (Wed-Fri)	GCTM Conference at Rock Eagle—a limited number of rooms are available for preservice teachers at $5.00 per night (GCTM membership required for this rate)
November 18-20 (Wed-Fri)	NCTM Regional Conference in Nashville
March 5-7 (Fri-Sun)	Teachers Teaching with Technology (T^3) International Conference in Atlanta –Registration ($145) is complimentary for Preservice Teachers—register by January 22.