EDCI 609 Problem Solving in Mathematics in the Middle Grades (3)

Section 001 (PDS) Fall (1997) T 11:00 - 4:00 Room A412

8/27, 9/16, 9/23, 9/30, 10/7, 10/21, 11/25, 12/2, 12/9

Instructor: Dr. Mark A. Spikell, Professor

Graduate School of Education Office: A318 Robinson

Phone: 993-2042 (office) 993-2013 (fax) mspikell@gmu.edu (e-mail)

Focuses on the learning processes fundamental to the development of mathematical thinking. A variety of instructional strategies and materials is examined and related to the broad scope of mathematical content. Field experience required.

Primary objectives of this course enable students to:

A. Understand and explain the steps in two models for solving mathematical problems -- Polya's 5-stage problem solving approach and the 7-stage scientific method approach.

B. Identify and apply these problem solving techniques:

C. Solve problems in the mathematical content areas of logic, geometry, algebra and number theory appropriate for adaptation to the middle grades.

Secondary objectives of this course provide students with:

D. Hands-on experience with selected manipulatives such as Color Cubes, Attribute Blocks, Geoboards, Cuisenaire Rods, Tangrams.

E. Various strategies and techniques for teaching problemsolving in the middle grades mathematics curriculum.

F. Alternative methods for assessing students' work in problem solving in the middle grades.

G. Experience working cooperatively to develop instructional materials and presentation skills to large and small groups.

1. Students purchase a math manipulatives kit at the bookstore consisting of: attribute logic blocks, snap cubes, Cuisenaire rods, geoboard, tangrams, a soma cube and Dr. Super=s Triangles
2. Required texts:

  A. Elementary School Mathematics: Teaching Developmentally, 2nd Ed., 1994, Van de Walle, John, Longman, 95 Church Street, White Plains, NY 10601.

B. Curriculum and Evaluation Standards for School Mathematics, 1989. National Council Teachers of Mathematics, 1906 Association Drive, Reston, VA 22091. 703-620-9840. (note: these texts will be usefull for this course and EDCI 528 Teaching and Learning Math in Middle Education. )

3. Some excellent additional optional resources are:

4. Optional mini subscription to Mathematics -- Teaching in the Middle School, a journal available from the National Council Teachers of Mathematics, 1906 Association Drive, Reston, VA 22091 703-620-9840.

5. Additionally, students may have access during the course to various handouts, materials, articles and specially designed activities and problem sets. (to be discussed in class)
6. Students must have access to email and the internet, either at home or at GMU. All GMU students receive an account providing them with these capabilities.

 Primary Resources Available in the George Mason Library:

1. Journals: Teaching Children Mathematics (formerly Arithmetic Teacher); Mathematics -- Teaching in the Middle School; Mathematics Teacher (MT); Journal for Research in Mathematics Education (JRME).

2. All Elementary School Mathematics Series Adopted by the Commonwealth of Virginia.

3. NCTM Curriculum and Evaluation Standards (closed reserve, GMU Library)

4. Mathematics Standards of Learning for Virginia Schools (closed reserve, GMU Library).

1. Fairfax County Professional Development Library, Fairfax High School, 3500 Old Lee Highway, 703-591-4514. Requires GMU Id.

2. National Council Teachers of Mathematics Library, 1906 Association Drive, Reston, VA 22091, 703-620-9840. Must be a member or student member to use.

3. Center for Interactive Educational Technology Labs R A350.

Examinations: (50%)

Projects: (40%)

1. (Required - 10%) Problem Solving Strategies Library Research Project -- Students will be assigned a problem solving strategy (see item B under course objectives). Students locate and read in appropriate professional journals two articles -- one article addressing in some way the assigned strategy, the other article dealing with any aspect of problem solving. Students prepare a 5-10 minute oral presentation to be given in class to small groups. The presentations should focus on (1) the central idea(s) in the articles; (2) explaining, clarifying, illustrating, etc. the strategy they were assigned; (3) sharing at least one non-routine problem (or activity), appropriate for use in the middle grades; and (4) relating the contents of the article or activity to one of the core NCTM standards other than problem solving (i.e., reaoning, communications or connections). The non-routine problem should illustrate how the assigned strategy is used to solve the problem. Appropriate handouts (e.g., copy of one or both articles) for each class member should be provided.
2. (Required - 20%) Technology/Manipulatives Project - Student 4-5 person teams will select, with the instructor=s approval, a specific manipulative or technology (one not explored in depth in class) that can be used in the middle grades to explore mathematical problem solving. Teams are responsible for (A) exploring the manipulative or technology, (B) utilizing it in some way in their internship and (C) preparing a small group activity center type presentation for the class that achieves the following goals: 1. Introduces classmates to the use of the manipulative or technology through hands-on activity; 2. Shows how the manipulative can be used in teaching problem solving in the middle grades; 3. Reports on the nature and results of the internship experience with the manipulative or technology. The oral presentation must include appropriate handouts summarizing the presentation=s content. Handouts must include a copy of at least two journal articles related to the presentation; one should discuss relevant research, theory, or philosophy; one should offer classroom activities or teaching suggestions.
3. (Required - 10%) Textbook Paper-Pencil Problem Solving Project -- Students will form their own teams of 4-5 persons to solve and prepare answer keys for these non-routine problems::

(A) the 55 problems (numbered 1 through 55) on pages 62 - 67 of the text, Helping Children Learn, and activity cards 4-1, 4-2, and 4-3 in Chapter 3 (page 59).

(B) the 57 problems from the text, Elementary School Mathematics: Teaching Developmentally. (See attachment)

(C) the 36 problems from the text, Problem Solving ... a basic mathematics goal (see attachment) in Challenge sets I, II, III, and !V and the Final Challenge Set

(D) the 20 problems from the text, How to Solve It. (See attachment)

 

 For each problem, present three labeled parts: I. Problem -- Clearly identify which problem is being solved (give the name, number or other identifying information). II. Work and Solution -- Once the problem is solved, show, in an organized fashion, all work done in arriving at the solution. III. Strategies -- Identify and name the key strategy(ies) used in solving the problem.

The required projects will be due as announced in class.

Reading and Problem Assignments: (All students must read these assignments by the time noted):

1. Required: Elementary School Mathematics: Teaching Developmentally, 2nd Ed.

(A) Chapter 1, Teaching Mathematics: Reflections and Directions. (class 3)

(B) Chapter 2, Doing Mathematics: Learning in A Mathematical Environment. (class 4)

(C) Chapter 3, Developing Understanding in Mathematics. (class 5)

(D) Chapter 4, Developing Problem-Solving Processes. (class 5)

(E) Chapter 5, Assessment in the Classroom. (class 6)

(F) Chapter 21, Technology and Elementary School Mathematics (class 7)

(G) Chapter 22, Planning for Developmental Instruction (class 7)

(H) Chapter 23 Mathematics and Children with Special Needs (class 8)

2. Required: Curriculum and Evaluation Standards for School Mathematics:

 

(A) Introduction (pg 1-12), by class 3

(B) K-4 Overview (pg 15-21), by class 5

(C) Grades 5-8 Overview (pg 65-73), by class 7

(D) Grades 9-12 Overview (pg 123-136) by class 9

 

3. Optional/Extra Credit: Teaching Problem Solving, What, Why & How, entire book.

 4. Problem Solving .. a basic mathematics goal: Becoming a Better Problem Solver, Bk 1, text material only.

For each optional/extra credit reading, submit a 1-2 page summary highlighting central ideas covered in the text. These are due at the final examination.

Course Grading: The final grade will be based on:

Problem Solving Strategy Project 10%

Manipulative/technology Project 20%

Paper-Pencil Problem Solving Project 10%

Attendance and Participation 10%

Final Exam 50%

 In the event of a borderline grade, class attendence and participation (oral in class and written outside of class) may be used in determining the final grade.

 Projects (oral or written), reading assignments and class participation are evaluated on a Satisfactory (A) or Unsatisfactory (C or F) basis using some of these criteria:

"How much of the assignment was completed correctly? ... Is the required information presented? ... Is the submission neat and well organized? ... Is it clearly written or spoken and presented in logical fashion? ... Are solutions and other commentary accurate? ... Does the student participate voluntarily in presenting problem solutions or speaking in discussions? ... Does the student participate actively in class explorations, discussions and homework assignments? ... Does the student attend class regularly?"

For projects and reading assignments, a Satisfactory evaluation is earned when oral and/or written presentations are complete, clear, well presented, thorough and accurate. For participation, a Satisfactory evaluation is earned when students consistently and accurately participate in class discussions, problem solving activities, problem reviews, small group work, etc.

Office Hours:

Dr. Spikell intends to be available before and after each class for up to one hour (and by appointment at other times) to accomodate students who wish assistence outside of class. Because of numerous meetings and other professional tasks, students should schedule meeting times in advance and not rely on published office hours to insure that they may have ample access for help on academic matters.

Tentative Course Outline (subject to modification):

The instructor reserves the right to change or modify the syllabus in response to emergency situations, the unexpected needs of students or for other appropriate purposes.

C1 .. Unit 1 Reasoning and Techniques for Teaching Problem Solving

Workshop I -- Attribute Blocks. Theme -- Developing Logical Thinking. Activities: (1) What's In The Box? (2) Two Ring Game; (3) Difference Trains; (4) Difference Matrices

C2 .. Unit 2 Generalizations, Proof and Models of Problem Solving

Workshop II -- Attribute Blocks. Theme -- Forming Generalizations and Role of Proof in the Curriculum: Activities: (1) I'm Thinking of a Block, (2) Extension (and Analysis) of the I'm Thinking of Something in the Room Activity 3) Sherlock's Block -- Discussion/Lecturette on Scientific Method and Polya Models of Problem Solving Workshop III Geoboards. Theme -- Teaching Area Concepts with a Problem Solving Approach: Activities, Designs, Copy Activity, Exploring Area (Interior/Exterior Methods)

C3 .. Unit 3 Connections -- Algebra/Geometry; NCTM Standards

Workshop IV -- Geoboards. Theme -- Relating Algebra and Geometry through Problem Solving; Activities: (1) Developing a Famous Theorem (2) Squares on a Geoboard. Theme -- National and State Guidelines for the Mathematics Curriculum; Activities: Lecture on (1) NCTM Standards (2) Virginia=s SOLs and (3) Rationale for Focus on Problem Solving in the School Math Curriculum

C4 ... Unit 4 Introduction to Spatial Visualization in 2 and 3 Dimensions

Workshop V -- Tangrams and Soma Puzzle. Theme -- Problem Solving and Spatial Visualization; Activities: Constructing 2 & 3 Dimensional Shapes, Perimeter, Area and Volume, Record Keeping Schema, Methods for Teaching Visualization via Problem Solving

C5 ... Unit 5 Developing Spatial Sense in 3 Dimensions

Workshop VI -- Polydrons. Theme -- Developing 3-D Spacial Sense; Activities: A. Spatial sense survey; B. Building polyhedra from nets; C.. Classifying polyhedra D. Building polyhedra from perspective sketches E. Relating problem solving, algebra and geometry -- discovering Euler=s formula

C6 ... Unit 6 Assessing Non-Routine Problem Solving

Workshop VII -- Cubes: Theme: Assessing Mathematical Problem Solving; Activity: The Architect=s Memo Problem; Lecture/Discussion on Analytic Methods for Assessing Non-Routine Problem Solving; Discussion on How to Determine A Best Answer for Non-Routine Problems

C7 .. Unit 7 Introduction to Strategy Games and the Role They Play in Problem Solving

Workshop VIII -- Counters: Theme -- Exploring Problem Solving through Strategy Games; Activities: Analysis of 3-In-A-Row Game, Jumping Frogs Puzzle; !5 Game; Line Game (optional); Others as Time Permits;

C8 .. Unit 8 Teaching Resources for Focusing on Problem Solving

Manipulative/Technology Presentations -- Overview of Other Available Resource Materials -- Presentation (by professor) on the Mathematics Embodied in the Various Manipulatives Demonstrated by Students

C9 .. Unit 8 Summarizing the Internship Experiences with Problem Solving and Final Exam

Discussions of Internship Experiences -- More on the Mathematics Embodied in the Various Manipulatives Demonstrated by Students -- Final Examination

Manipulative/Technology Project Choices:

(1) 4-function or scientific calculators to teach arithmetic and problem solving;

(2) graphing calculators to teach algebra and data analysis;

(3) computer software, the INTERNET, and video technology to teach problem solving, probability or statistics, algebra or geometry;

(4) polydrons to teach geometry;

(5) pattern blocks to teach arithmetic and geometry;

(6) hundreds boards to teach arithmetic and problem solving;

(7) algebra lab gear to teach algebra;

(8) fraction bars to teach fractions and fraction operations;

(9) Dr. Super=s triangles to teach fractions, algebra or symmetry;

(10) decimal squares to teach decimals and decimal operations;

(11) Mortensen math materials to teach selected mathematics topics;

(12) polyhedra dice to teach probability and statistics;

(13) powers of 10 blocks to teach arithmetic and geometry;

(14) Cuisenaire rods to teach arithmetic and algebra;

(15) tangrams to teach perimeter and area;

(16) soma cubes to teach spatial sense;

(17) geoboards to teach measurement and geometry;

(18) snap cubes or linking cubes to teach spatial sense in 3-dimensions, algebra, and geometry;

(19) color cubes to teach strategy game analysis, arithmetic, algebra and geometry;

(20) dime solids to teach spatial sense in 3-dimensions;

(21) algeblocks

Suppliers of Manipulative Materials in Mathematics Education:

1. ETA (Educational Teaching Aids) 1-800-445-5985
2. Cuisenaire Company of America 1-800-237-0338
3. Dale Seymour Publications 1-800-872-1100
4. Creative Publications 1-800-624-0822
5. Scott Resources 1-800-446-8767
6. Stokes Publishing Company 1-408-541-9145 (1292 Reamwood Ave, Sunnyvale, CA 94089)