Mathematics Assessment What Works in the Classroom

by Kulm, Gerald

• ISBN13: 9780787900403
• ISBN10: 0787900400
• Edition: 1st
• Format: Paperback
• Copyright: 1994-12-13
• Publisher: Jossey-Bass

Summary

"A thoughtful, idea-filled book for educators confronting the challenge of creating and using alternative assessment procedures. Drawing on years of work with mathematics teachers, Kulm turns the focus on what really works in the classroom. A must read for teachers looking for practical suggestions and potential models as they develop new assessment systems." -- Thomas A. Romberg, director, National Center for Research in Mathematical Sciences Education, University of Wisconsin-MadisonKulm offers hands-on guidelines for assessing student understanding of procedures, concepts, and problem-solving, including advice on developing and interpreting assessment tasks, assessing individual and group mathematical activities, using scoring rubrics, developing student portfolios, and more.

Author Biography

GERALD KULM is director of evaluation for the STEP Mathematics and Science Teacher Preparation Project at Montana State University. He was director of mathematics and science education programs at the American Association for the Advancement of Science.

Table of Contents

Preparing to Teach Mathematics	1	(15)
Philosophical Considerations	4	(1)
What Is Mathematics?	4	(1)
What Does It Mean to Know Mathematics?	4	(1)
Implications of These Philosophical Considerations	5	(1)
Influences on the School Mathematics Curriculum	5	(3)
Professional Organizations	5	(1)
Principles and Standards for School Mathematics	6	(1)
Mathematics Textbooks	6	(1)
Assessment Procedures	7	(1)
State Governmental Bodies	8	(1)
Looking Ahead	8	(2)
NCTM Standards	8	(1)
Developing Skills	8	(1)
The Math Program	9	(1)
Communicating	9	(1)
Connecting	9	(1)
Assessing Mathematics Learning	9	(1)
Something for Everyone	10	(1)
Teaching Considerations	10	(2)
Key Ideas	12	(1)
Discussion Questions	12	(1)
References	12	(3)
Children and Mathematics	15	(20)
A Constructivist View of Learning	18	(1)
The Children's World	19	(4)
Children Have Many Number Experiences	19	(1)
Children Are Active in Their World	20	(1)
Children Observe Relationships in Their World	20	(1)
Children Learn Mathematics in Concert with Other Subjects	20	(1)
Children's Feelings Affect Their Ability to Learn	21	(2)
Psychological Considerations in Teaching Mathematics	23	(6)
Logical and Psychological Approaches to Mathematics	23	(1)
Sources of Information about How Children Learn Mathematics	24	(1)
How Children Form Mathematical Concepts	25	(1)
Dealing with Concepts Once Formed	26	(1)
Children's Thinking	26	(1)
Children's Communication of Mathematical Concepts	27	(2)
Teaching Children Mathematics	29	(2)
Teachers' Attitudes about Children	29	(1)
Teaching Style	29	(1)
Enhancing and Guiding Learning	30	(1)
Key Ideas	31	(1)
Discussion Questions	32	(1)
References	32	(3)
Solving and Creating Problems, Estimating, and Mental Calculating	35	(28)
Problem Solving	38	(11)
What Is a Problem?	38	(1)
Solving and Creating Problems	39	(1)
Characteristics of Good Problem Solvers and Good Problem Creators	40	(1)
Influences on a Person Trying to Solve or Create Problems	40	(1)
Heuristics or Strategies Used in Solving and Creating Problems	41	(7)
Problem Solving in Projects	48	(1)
Estimating	49	(1)
Weekly Estimations	49	(1)
Mental Calculating	50	(2)
Warm-Ups for a Math Lesson	50	(2)
Grouping Students for Problem Solving, Estimating, and Mental Calculating	52	(1)
Communicating in Problem Solving	53	(1)
Connecting Learning of Problem Solving, Estimating, and Mental Calculating	53	(1)
Assessing Learning of Problem Solving, Estimating, and Mental Calculating	53	(6)
Something for Everyone	59	(1)
Key Ideas	59	(1)
Discussion Questions	59	(1)
References	60	(3)
Technology in Teaching and Learning Mathematics	63	(18)
Calculators	66	(5)
How Should Calculators Be Used?	67	(3)
What Types of Calculators Should Be Purchased?	70	(1)
Computers	71	(6)
Other Technology	77	(1)
Overhead Projectors	77	(1)
Lettering Machines	77	(1)
Videotapes and Cameras	78	(1)
Slides and Cameras	78	(1)
Interactive Video, Telecommunications, and Other Newer Technologies	78	(1)
Key Ideas	78	(1)
Discussion Questions	78	(1)
References	79	(2)
Thinking Mathematically	81	(36)
Developing Concepts	85	(11)
Observing and Inferring	85	(2)
Comparing	87	(1)
Classifying	87	(6)
Sequencing	93	(1)
Relationships	93	(3)
Developing and Practicing Skills	96	(9)
Carroll Diagrams	96	(3)
Tree Diagrams	99	(3)
Venn Diagrams	102	(3)
Estimating	105	(2)
Solving and Creating Problems	107	(3)
Grouping Students for Mathematical Thinking	110	(1)
Communicating Learning of Mathematical Thinking Processes	110	(1)
Connecting Learning of Mathematical Thinking	111	(1)
Assessing Learning of Mathematical Thinking	111	(3)
Something for Everyone	114	(1)
Key Ideas	114	(1)
Discussion Questions	115	(1)
References	115	(2)
Attaching Meaning to Numbers	117	(34)
Developing Number Concepts	121	(10)
Conservation	122	(2)
Combinations	124	(1)
Zero	124	(1)
One-to-One Correspondence and Counting	125	(1)
Measuring Length	125	(2)
Grouping	127	(4)
Developing and Practicing Number Skills	131	(9)
Matching Numerals to Sets	131	(1)
Writing Numerals	132	(1)
Rational Counting	132	(3)
Number Names	135	(1)
Place Value	135	(5)
Estimating	140	(1)
Solving and Creating Problems	141	(4)
Grouping Students for Learning the Meaning of Numbers	145	(1)
Communicating Understanding of the Meaning of Numbers	145	(1)
Connecting Number Learning	146	(1)
Assessing Number Learning	146	(1)
Something for Everyone	147	(1)
Key Ideas	147	(1)
Discussion Questions	148	(1)
References	148	(3)
Teaching and Learning Addition and Subtraction	151	(40)
Developing Addition and Subtraction Concepts	155	(9)
Operations	156	(1)
Addition	157	(4)
Subtraction	161	(3)
Developing and Practicing Addition and Subtraction Skills	164	(13)
Basic Addition and Subtraction Facts	165	(5)
Addition and Subtraction Algorithms	170	(7)
Estimating and Mental Calculating	177	(3)
Solving and Creating Problems	180	(6)
Grouping Students for Learning Addition and Subtraction	186	(1)
Communicating Learning of Addition and Subtraction	186	(1)
Connecting Learning of Addition and Subtraction	186	(1)
Assessing Learning of Addition and Subtraction	187	(1)
Something for Everyone	188	(1)
Key Ideas	188	(1)
Discussion Questions	189	(1)
References	189	(2)
Teaching and Learning Multiplication and Division	191	(38)
Developing Multiplication and Division Concepts	194	(7)
Multiplication	195	(3)
Division	198	(3)
Developing and Practicing Multiplication and Division Skills	201	(17)
Basic Multiplication and Division Facts	201	(5)
Multiplication and Division Algorithms	206	(12)
Estimating and Mental Calculating	218	(2)
Solving and Creating Problems	220	(3)
Grouping Students for Learning Multiplication and Division	223	(1)
Communicating Learning of Multiplication and Division	223	(1)
Connecting Learning of Multiplication and Division	224	(1)
Assessing Learning of Multiplication and Division	225	(1)
Something for Everyone	226	(1)
Key Ideas	227	(1)
Discussion Questions	227	(1)
References	228	(1)
Attaching Meaning to Rational Numbers	229	(32)
Developing Rational Number Concepts	233	(10)
Part-Whole Model for a Common Fraction	233	(3)
Ratio Model for a Common Fraction	236	(2)
Using Common Fractions to Indicate Division	238	(1)
Decimal Fractions	239	(2)
Percent	241	(1)
Equivalent Fractions	241	(1)
Ordering Fractions	242	(1)
Developing and Practicing Rational Number Skills	243	(8)
Renaming Common Fractions	243	(3)
Reading and Writing Decimal Fractions	246	(1)
Renaming Common Fractions, Decimal Fractions, and Percents	247	(1)
Ordering Common and Decimal Fractions	248	(1)
Proportions	249	(2)
Estimating and Mental Calculating	251	(1)
Solving and Creating Problems	252	(1)
Grouping Students for Learning Rational Number Concepts	253	(1)
Communicating Learning of Rational Number Concepts	253	(3)
Connecting Learning of Rational Numbers	256	(1)
Assessing Learning of Rational Numbers	256	(2)
Something for Everyone	258	(1)
Key Ideas	258	(1)
Discussion Questions	258	(1)
References	259	(2)
Teaching and Learning Operations with Rational Numbers	261	(42)
Developing Concepts of Operations with Rational Numbers	265	(22)
Addition of Common Fractions	265	(2)
Addition of Decimal Fractions	267	(4)
Subtraction of Common Fractions	271	(2)
Subtraction of Decimal Fractions	273	(1)
Multiplication of Common Fractions	274	(3)
Multiplication of Decimal Fractions	277	(2)
Multiplication with Percent	279	(1)
Division of Common Fractions	280	(3)
Division of Decimal Fractions	283	(4)
Developing and Practicing Skills in Operations with Rational Numbers	287	(5)
Addition and Subtraction of Common Fractions	287	(2)
Addition and Subtraction of Decimal Fractions	289	(1)
Multiplication and Division of Common Fractions	290	(1)
Multiplication and Division of Decimal Fractions	291	(1)
Operations with Percents	292	(1)
Estimating and Mental Calculating with Rational Number Operations	292	(2)
Solving and Creating Problems	294	(1)
Grouping Students for Learning Rational Number Operations	295	(1)
Communicating Learning of Rational Number Operations	296	(1)
Connecting Learning of Rational Number Operations	296	(1)
Assessing Learning of Rational Number Operations	297	(1)
Something for Everyone	297	(2)
Key Ideas	299	(1)
Discussion Questions	299	(1)
References	300	(3)
Teaching and Learning Probability, Statistics, and Graphing	303	(28)
Developing Concepts of Probability, Statistics, and Graphing	306	(12)
Probability Concepts	306	(3)
Statistics Concepts	309	(5)
Graphing Concepts	314	(4)
Developing Skills in Probability and Graphing	318	(3)
Probability Skills	318	(2)
Graphing Skills	320	(1)
Estimating and Mental Calculating	321	(1)
Solving and Creating Problems	322	(2)
Grouping Students for Learning Probability, Statistics, and Graphing	324	(1)
Communicating Learning of Probability, Statistics, and Graphing	324	(1)
Connecting Learning of Probability, Statistics, and Graphing	325	(1)
Assessing Learning of Probability, Statistics, and Graphing	325	(1)
Something for Everyone	326	(1)
Key Ideas	327	(1)
Discussion Questions	327	(1)
References	327	(4)
Teaching and Learning Geometry	331	(50)
Developing Geometric Concepts	335	(27)
Young Children's Views of the World	337	(2)
Projective Geometry	339	(2)
Plane Figures and Their Properties	341	(8)
Symmetry and Motion	349	(6)
Space Figures	355	(3)
Fractal Geometry	358	(4)
Developing and Practicing Geometric Skills	362	(6)
Identifying Shapes	362	(1)
Copying and Inventing Shapes	362	(1)
Constructing Shapes	363	(5)
Estimating	368	(2)
Solving and Creating Problems	370	(5)
Grouping Students for Learning Geometry	375	(1)
Communicating Learning of Geometry	376	(1)
Connecting Learning of Geometry	376	(1)
Assessing Learning of Geometry	377	(1)
Something for Everyone	378	(1)
Key Ideas	378	(1)
Discussion Questions	378	(1)
References	378	(3)
Teaching and Learning Measurement	381	(40)
Developing Measurement Concepts	384	(7)
Length and Distance	385	(1)
Area	386	(1)
Weight and Mass	387	(1)
Time	388	(1)
Volume and Capacity	389	(1)
Temperature	389	(1)
Angle	390	(1)
Developing and Practicing Measurement Skills	391	(23)
Measuring Length	391	(6)
Measuring Area	397	(3)
Measuring Weight and Mass	400	(3)
Measuring Time	403	(2)
Measuring Volume and Capacity	405	(3)
Measuring Temperature	408	(2)
Measuring Angles	410	(4)
Estimating	414	(1)
Solving and Creating Problems	414	(2)
Grouping Students for Learning Measurement	416	(1)
Communicating Learning of Measurement	417	(1)
Connecting Learning of Measurement	417	(1)
Assessing Learning of Measurement	418	(1)
Something for Everyone	418	(1)
Key Ideas	419	(1)
Discussion Questions	419	(1)
References	420	(1)
Introducing Algebra	421	(28)
Developing Algebraic Concepts	425	(16)
Patterns and Relationships	425	(2)
Integers and Their Operations	427	(3)
Functions, Variables, Equations, and Inequalities	430	(9)
Solving Equations and Inequalities	439	(2)
Developing and Practicing Algebraic Skills	441	(1)
Estimating and Mental Calculating	442	(1)
Solving and Creating Problems	442	(1)
Grouping Students for Algebraic Learning	443	(1)
Communicating Learning of Algebraic Skills	443	(1)
Connecting Learning of Algebraic Concepts	443	(1)
Assessing Learning of Algebraic Concepts and Skills	444	(1)
Something for Everyone	444	(1)
Key Ideas	445	(1)
Discussion Questions	445	(1)
References	446	(3)
Organizing for Mathematics Instruction	449	(28)
Fitting Instruction to Students' Differing Needs and Styles	452	(4)
Planning for Teaching	453	(1)
Cooperative Learning	454	(2)
Assessing, Recording, and Reporting Students' Progress	456	(6)
Assessing Students' Progress	457	(4)
Recording Students' Progress	461	(1)
Reporting Students' Progress	461	(1)
Children with Special Needs	462	(3)
Children with Special Problems	462	(1)
Children with Special Mathematical Talents	463	(2)
Arranging the Classroom	465	(4)
Providing Learning Aids	469	(5)
Commercial Materials	469	(2)
Teacher-Made Learning Aids	471	(3)
Student-Made Learning Aids	474	(1)
Constructing Learning Aids	474	(1)
Key Ideas	474	(1)
Discussion Questions	475	(1)
References	475	(2)
Appendix A Suppliers of Manipulative Materials and Computer Software	477	(4)
Appendix B Blackline Masters	481	(33)
Index	514