Type:
Solution Manual
Resource:
Helping Children Learn Mathematics
Edition:
10th Edition
Author(s):
Robert E. Reys Mary Lindquist Diane V. Lambdin Nancy L. Smith
Instructor’s Resource Guide, Chapter 1 Sandi Cooper |1
Chapter 1 —School Mathematics in a Changing World Chapter Overview In this chapter, the student is asked to examine what mathematics is and what determines the mathematics being taught. A brief historical perspective of the mathematics curriculum is discussed with a clear presentation of the underlying Principles developed by the National Council of Teachers of Mathematics [NCTM]. Three general factors (needs) are presented as influences on the mathematics curriculum. The chapter concludes with a discussion of resources teachers can use to learn more about teaching mathematics. Student Objectives After reading the chapter, the students will be able to: 1.
Describe five views of mathematics.
2.
Summarize three needs that influence the mathematics curriculum.
3.
Summarize recommendations from NCTM’s six underlying Principles of school mathematics.
4.
Identify resources for learning more about mathematics teaching.
Key Vocabulary For this chapter, many of the key terms involve connections to names of persons and organizations. Students may have encountered these names and terms in earlier courses with others being specifically tied to the mathematics subject area. Journal for Research in Mathematics Education [JRME] learning Teaching Children Mathematics [TCM] assessment Mathematics Teaching in the Middle School [MTMS] technology Principles and Standards for School Mathematics [PSSM] new math field theory (Gestalt) Committee of Seven meaning and understanding (Brownell) teaching mathematical proficiency (NRC) equity social utility (Wilson) curriculum connectionism (Thorndike) construct knowledge National Council of Teachers of Mathematics [NCTM] mental discipline National Assessment of Educational Progress [NAEP] incidental learning Trends in Mathematics and Science Study [TIMSS] No Child Left Behind [NCLB] Common Core State Standards Initiative [CCSSI] .S o n s
Instructor’s Resource Guide, Chapter 1 Sandi Cooper |2 Supplemental Lecture Topics 1.
Provide more background information about NCTM's Principles and Standards for School Mathematics, Professional Standards for Teaching Mathematics, Assessment Standards for School Mathematics and/or Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics. The booklet, Principles and Standards for School Mathematics: An Overview, the Principles and Standards Outreach CD 2nd Ed., the Quick Reference Guide: Table of Standards and Expectations, and Answers to Frequently Asked Questions about PSSM, and A Research Companion to Principles and Standards for School Mathematics are available from NCTM and provide useful information.
2.
Provide information concerning NCTM's student memberships and starter kits for students. NCTM has packets of information that can be distributed to students. The Member Services and Public Relations Department may be reached at (703)620-9840 ext. 113. Have students examine sample copies of the NCTM Educational Material Catalog, Teaching Children Mathematics (formerly Arithmetic Teacher), Mathematics Teaching in the Middle School, Mathematics Teacher, and Journal for Research in Mathematics Education (JRME).
3.
Discuss mandates and recommendations for mathematics education that are currently being enacted in your state or region. For example, implementation of the Common Core State Standards, state goals, grade level objectives, state assessment instruments, NCLB,etc. Discuss the influence these mandates have on classroom instruction. Invite a practicing teacher to discuss his/her view of initiatives.
4.
Share results from the Trends in Mathematics and Science Study (TIMSS), from the books, Standards & Curriculum: A View from the Nation, Results and Interpretations of the 1990 through 2000 Mathematics Assessments of the National Assessment of Educational Progress, or Mathematics Education in the United States 2004.
5.
The NCTM 2000 Yearbook, Learning Mathematics for a New Century and the 2004 Yearbook, Perspectives on the Teaching of Mathematics, includes several timely articles about issues in mathematics education. Use these as discussion starters or assign various students to read and report to the class.
6.
Implementing Standards-Based Math Instruction: A Casebook for Professional Development and Reflecting on NCTM’s Principles and standards in Elementary and Middle School Mathematics: Readings from NCTM’s School-Based Journals are available from NCTM. Thinking like Mathematicians: Putting the NCTM Standards into Practice is available from Heinemann. They provide research-based cases articles that illustrate reform efforts and ideas for group discussions. Use these cases or articles to help students envision how reform occurs in classrooms.
7.
Share more information on school mathematics history from A History of School Mathematics, Volumes I and II.
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Instructor’s Resource Guide, Chapter 1 Sandi Cooper |3 Student Textbook Activities 1.
As an advance organizer, have students read the Focus Questions, found at the beginning of the chapter and on PPT Slide #2. Discuss what they already know and what they want to learn more about. Have them share any other questions they have about this chapter.
2.
Involve students in a discussion about what mathematics is, what it has been and what it should be. These questions are central to the discussion of the mathematics curriculum and the forces affecting its form. Most prospective teachers and elementary children, based on their past experiences, may tend to view mathematics as arithmetic. Mathematics is often viewed as a quest for the "correct answer". Have students examine Figure 1-1, Vision of the mathematics content distribution through the grade levels and have them relate this to their own school experiences.
3.
PPT Slide #3, What is Mathematics? describes mathematics in five different ways - as discussed in the text. You may want to start by having students compile a list of words they associate with mathematics. Using these, lead the discussion to the five views of mathematics and have the class brainstorm examples that fit within each definition.
4.
PPT Slides #6-12, Principles for School Mathematics, provides the five underlying principles of school mathematics as outlined in NCTM’s Principles and Standards for School Mathematics. These may also be found in the text on pages 6-7. Encourage students to discuss each principle and its implications for them as future elementary teachers.
5.
In small groups, have students develop a vertical timeline (1900, 1910...1990, 2000). Have each group locate key items from chapter 1 and place them on the timeline. Discuss the results as a class. A suggested timeline follows which may be cut apart for groups to locate in the text. Follow up with a discussion of PPT Slides #13-14, Major Trends in School Mathematics.
Late 1800's 1900 1920's 1930's 1930's 1931 1940s 1957 1950's, 60's 1960s 1970's 1973 1980’s .S o n s
Mental Discipline Thorndike-connectionism progressive movement, incidental learning Gestalt-field theory Brownell-meaning and understanding Committee of Seven Wilson-social utility Sputnik New Math Movement concern for disadvantaged students Minimal-Competency Movement NAEP begins Problem Solving Movement
Instructor’s Resource Guide, Chapter 1 Sandi Cooper |4 1989 1990s 1991 1995 1996 2000 2000 2001 2001 2003 2006 2010
NCTM's Curriculum and Evaluation Standards for School Mathematics Standards movement NCTM's Professional Standards for Teaching Mathematics NCTM's Assessment Standards for School Mathematics TIMSS results NAEP results from the Seventh Mathematics Assessment released NCTM's Principles and Standards for School Mathematics released Adding it Up released by the National Research Council No Child Left Behind Trends in Mathematics and Science Study (TIMSS) NCTM’s Curriculum Focal Points for Grades K-8 Common Core State Standards Initiative
7.
Use PPT Slides #15-16, Resources for Teachers, to discuss the suggestions provided in the text. Encourage students to share specific examples for each that they have seen or used.
8.
The Math Links in this chapter provide students with additional resources for this chapter. In class, or as an outside assignment, have students explore these web sites.
9.
Provide students with the opportunity to discuss unresolved concerns about the elementary school mathematics curriculum and preview the upcoming chapters where the identified concerns will be addressed in greater detail. Use the following discussion starters to stimulate dialogue and encourage brainstorming in small groups. Starter #1: Brainstorm concerns which will need to be addressed by the mathematics education community as curriculum change is implemented. Starter #2: Brainstorm what classroom teachers can do as they prepare to change the elementary mathematics curriculum.
Student Supplemental Activities 1.
.S o n s
Have students complete an introductory survey on the first day of class. These surveys may be returned to students on the last day of class and changes that occurred may be discussed. The survey may include items such as: • Math is... • When it comes to math, I feel... • I think my math ability is... • Elementary math should be... • It is most important that elementary math students learn... • Name one thing you remember from your math classes in elementary school. • Describe what you hope to learn from this course. • How I feel about teaching mathematics is…
Instructor’s Resource Guide, Chapter 1 Sandi Cooper |5 2.
As students introduce themselves, have them share information from one of their survey items such as a memory (good or bad) from their math classes in elementary school. Discuss implications for their future teaching of mathematics or preview how the items discussed will be addressed in the course.
3.
Present to the students a list of myths about mathematics as presented in the article by Martha L. Frank, What Myths about Mathematics are Held and Conveyed by Teachers? (Arithmetic Teacher, 37(5): 10-12 - Jan 1990). Allow them to discuss in small groups and then share as a whole group. Discuss how certain beliefs might be conveyed unintentionally through instruction.
4.
Start a collection of magazine and newspaper articles related to mathematics education, student performance, the accountability or reform movement. For example, titles such as "Study: U.S. Students poor in math and science", or "U.S. Students Math Scores Up"... Divide students into groups to discuss the articles and then summarize results for the class. Start a display and encourage students to bring in articles throughout the course.
5.
Have the class build "get acquainted" line plots or graphs using materials such as grid paper, Post-It Notes®, links or paper clips, or connecting cubes (See Figures 17-3 and 17-4 in Chapter 17). Prior to collecting the data, have students discuss and predict the outcomes. You may have students estimate the number of candies in a jar or respond to questions such as: • How do you feel about teaching mathematics? Stack your cube under the happy face, so-so face, or sad face. • What grade do you most want to teach? Hang your link on your favorite grade. • What month were you born? Place your Post-it Note® under the correct month. • What is the first letter of your first name? Mark an x on the line plot above the letter. • How many letters in your first name? Estimate the average number of letters in first names in this class. Examine the variety of ways elementary students can easily display data. Have students brainstorm other questions which could be used as get acquainted mathematics activities with elementary students such as name your favorite color, measure your height in centimeters...
6.
Have students work in groups to discuss items from Things to do: From What You've Read, Things to do: Going Beyond this Book (page 11).
Field Experiences Additional activities, suggestions, and questions for students to complete in a school field experience are provided in the companion book, Teaching Elementary Mathematics: A Resource for Field Experiences, 4th edition. The following activities have been designed to be used with Chapter 1: .S o n s
Instructor’s Resource Guide, Chapter 1 Sandi Cooper |6 1.
Learning about the School and Its Resources: Curriculum Guide, Mathematics in the School
2.
Interviewing the Teacher and Students: Interviewing the Teacher about Equity
Additional Resources Adding It Up: Helping Children Learn Mathematics. National Research Council, National Academy Press, by Jeremy Kilpatrick, Jane Swafford, and Bradford Findell. Shares recommendations drawn from research and defines mathematical proficiency. Answers to Frequently Asked Questions about Principles and Standards for School Mathematics. Provides answers to commonly asked questions about PSSM. Available from NCTM. A History of School Mathematics, edited by George M.A. Stanic and Jeremy Kilpatrick, provides a record of the history of mathematics education in the United States and Canada. Available from NCTM. Implementing Standards-Based Mathematics Instruction: A Casebook for Professional Development, by Mary Kay Stein, Margaret Schwan Smith, Marjorie A. Henningsen, and Edward A. Silver. Shares cases and materials for use in classes and workshops. Available from Teachers College Press and NCTM. Learning Mathematics for a New Century (2000 Yearbook), provides several articles about issues in mathematics education. Available from NCTM. Lessons Learned from Research. Edited by Judith Sowder and Bonnie Schapelle. This book shares research in a teacher friendly format. Available from NCTM. Making the Grade in Mathematics: Elementary School Mathematics in the United States, Taiwan, and Japan--By Harold W. Stevenson, Max Lummis, Shin-Ying Lee, and James W. Stigler. Compares mathematics performance of elementary school students. Focuses on the differences in mathematics teaching methods and attitudes. Available from NCTM. Mathematics Education in the United States 2004, edited by Zalman Usiskin and John A. Dossey, is a report from the Tenth International Congress on Mathematical Education. Available from NCTM. An Overview of Principles and Standards for School Mathematics, A booklet which presents the main ideas that are presented in the latest Standards document. Available from NCTM. Perspectives on the Teaching of Mathematics (2004 Yearbook), focuses on teaching mathematics. Available from NCTM. Principles and Standards for School Mathematics Outreach CD, 2nd ed. Contains Powerpoint presentations, handouts, and video clips. Available from NCTM. .S o n s