Principles and Standards for School Mathematics How Were They Developed? How Can We Use Them? Glenda Lappan Metropolitan Mathematics Club November 17, 2000
Reusser (1986) There are 125 sheep and 5 dogs in a flock. How old is the shepherd? 3 out of every 4 students give an answer. “125 + 5 = 130…This is too big…while 125-5 = 120 is still too big…while 125/5 = 25. That works. I think the shepherd is 25 years old.” Math is arbitrary, useless, and meaningless.
What is this reform about? In a nutshell: • a modernization of the curriculum, • improved classroom instruction, • and assessment of student progress that informs and supports the continued mathematical learning of each student.
We need programs that are: • Developmentally responsive • Academically excellent • Socially equitable
Accommodating Students: • Connecting to students’ interests • Looking at the logical and experimental sides of mathematics • Creating a tool-rich environment for learning mathematics • Listening to our students to understand what sense they are making of mathematics
The Process of Developing Principles and Standards • Who was involved? • Teachers • School administrators • Mathematics supervisors • University mathematicians • Mathematics educators • Researchers
Soliciting Reactions Data Analysis(Coding the Responses)
Challenges • Qualitative nature of the data • Multiple purposes for analysis • To support development of issues • To capture specific content and editorial comments
Initial Decisions • General methodology • Use of NUD*IST software • Unit of analysis
Categorization Scheme General information • Category (1 x) Source of input • Category (2 x) Demographics Location in draft • Category (3 x) Chapter • Category (4 x) Principles • Category (5 x) Standards 11
Categorization Scheme (cont.) Type of reaction • Category (6 x) Editorial or stylistic • Category (7 x) Substantive concern Nature of reaction • Category (8 x) 11
Sample Coding [p.17] Line 26. This is a strong section. I'd lead off with this! To me as a teacher and curriculum writer, the most important role of the Standards is "expert" opinion on what's important to teach, something teachers stuck in schools all day cannot possibly get a good perspective on. What grounds the standards are a critical part for me! Unfortunately, you say "research cannot select the standards" and that what to teach is a value judgment. It's unfortunate that the NCTM did not generally base its standards on research about what math is really needed "out there.” (1 2) /Source/ARG report (2 2) /Who/Mathematician (3 1) /Chapter/Chapt. 1 (7 11) /Substantive Issues/Audience; purpose (7 21) /Substantive Issues/Research base (8 2) /Judgments/Effective message (8 4) /Judgments/Wrong message
Issue Clusters • Over-arching Issues (6) • Structure of Document (5) • Content Issues (4) • Issues Related to Learning (2) • Issues Related to Equity (2)
Overarching Issues 1. Audience and purpose 2. Specificity 3. View of mathematics 4. Relation to previous standards 5. Vision 6. Support of teachers
Structure of Document 7. Holistic view of classroom instruction 8. Role of chapter 3 9. Use of research 10. Role of examples 11. Readability and terminology
Content Issues 12. Connections within mathematics 13. Articulation across grades 14. Appropriateness of content for grade bands 15. Influence of technology
Issues Related to Learning 16. Use of particular theories 17. Balance of skills and understanding
Issues Related to Equity 18. Special student populations 19. Equity
Conference Board on the Mathematical Sciences American Mathematical Association of Two-Year Colleges American Mathematical Society Association of Mathematics Teacher Educators American Statistical Association Association for Symbolic Logic Association of State Supervisors of Mathematics Association for Women in Mathematics Benjamin Banneker Association Institute of Mathematical Statistics Mathematical Association of America National Association of Mathematicians National Council of Supervisors of Mathematics Society for Industrial and Applied Mathematics “With this, NCTM has established a model, heretofore all too rare, of how to stage civil, disciplined, and probing discourse among diverse professionals on matters of mathematics education.” (p.xv)
National Research Council: The committee finds that the process established by the NCTM to solicit comments from the field to be commendable and the process established by them to analyze those comments exemplary. The CRC also finds the NCTM’s response to the issues identified from those comments to be both adequate and appropriate. (May, 2000)
NCTM Presents “Higher Standards for Our Students... Higher Standards for Ourselves”
Principles and Standards for School Mathematics • A comprehensive and coherent set of goals for improving mathematics teaching and learning in our schools. • A vision around which to have focused conversations about improving mathematics programs. 22
Document Structure • A Vision for School Mathematics • Principles for School Mathematics • Standards for School Mathematics • FOUR GRADE-BAND CHAPTERS: Standards for… pre-K–2, 3–5,6–8, 9–12 • Moving toward the Vision • Table of Standards and Expectations
Features • mathematical tasks • illustrations of the use of technology in teaching, e-examples • student work • classroom episodes • research references
Teaching Assessment Technology Principles Describe particular features of high-quality mathematics programs • Equity • Curriculum • Learning
The Curriculum Principle A curriculum is more than a collection of activities: it must be coherent, focused on important mathematics, and well articulated across the grades. The Teaching Principle Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well. Statement of Principles The Equity Principle Excellence in mathematics education requires equity– high expectations and strong support for all students.
The Assessment Principle Assessment should support the learning of important mathematics and furnish useful information to both teachers and students. The Technology Principle Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students’ learning. Statement of Principles The Learning Principle Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge.
Choose a Principle • What would the principle look like in action? • How do your programs stack up? • What can you do to move toward the principle?
Number and Operations Algebra Geometry Measurement Data Analysis and Probability Problem Solving Reasoning and Proof Communication Connections Representation The Standards
Number and Operations Standard Instructional programs from prekindergarten through grade 12 should enable all students 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
Number and Operations Standard: • Compute fluently and make reasonable estimates • Develop fluency with basic number combinations for addition and subtraction Pre-K-2 • Develop fluency in adding, subtracting, multiplying, and dividing whole numbers 3-5 • Develop and analyze algorithms for computing with fractions, decimals, and integers and develop fluency in their use 6-8 • Develop fluency in operations with real numbers, vectors, and matrices, using mental computation or paper-and-pencil calculations for simple cases and technology for more-complicated cases 9-12 33
Geometry Standard Instructional programs from prekindergarten through grade 12 should enable all students to— • Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships; • Specify locations and describe spatialrelationships using coordinate geometry and other representational systems; • Apply transformations and use symmetry to analyze mathematical situations; • Use visualization, spatial reasoning, and geometric modeling to solve problems.
Geometry Standard: • Use visualization, spatial reasoning, and geometric modeling to solve problemsof two- and three-dimensional geometric shapes and • Recognize and represent shapes from different perspectives; Pre-K-2 • Identify and build a three-dimensional object from two-dimensional representations of that object; 3-5 • Use two-dimensional representations of three-dimensional objects to visualize and solve problems; 6-8 • Visualize three-dimensional objects and spaces from different perspectives and analyze their cross sections. 9-12 35
Reasoning and Proof Standard Instructional programs from prekindergarten through grade 12 should enable all students to— • Recognize reasoning and proof as fundamental aspects of mathematics • Make and investigate mathematical conjectures • Develop and evaluate mathematical arguments and proofs • Select and use various types of reasoning and methods of proof
Number Emphasis Across the Grades Pre-K–2 3–5 6–8 9–12 Algebra Geometry Measurement Data Analysis and Probability
Ron Richhart “Providing new kinds of learning opportunities that allow students to demonstrate their mathematical talent in broad and meaningful ways can very positively affect the expectations of teachers and students about what constitutes mathematical ability.”
A Vision of School Mathematics: Examples 39
Pre-K–2 Throughout the early years, the Standards can help parents and educators give children a solid foundation in mathematics. 40
Computational Fluency • Using efficient, accurate, and flexible methods for computing • Developing sensible strategies based on well-understood properties and number relationships • Having fluency with basic number combinations • Using estimation when calculating mentally, with paper and pencil, or with a calculator
A Vision of School Mathematics: E-standards Examples Grades K-2 4.2 Triangles and Polygons Grades K-2 4.3 Hiding Ladybugs Grades K-2 4.4 Tangram Puzzles 46
Grades 3–5 Nearly three-quarters of U.S. fourth graders report liking mathematics. 47
I thought seven 25’s - that’s 175. Then I need seven 3’s or 21. So the answer is 175 + 21 = 196 7 x 20 is 140 and 7 x 8 is 56 56 + 140 is 196 7 x 28 I did 7 x 30 first. That’s 210. Then take off seven 2’s or 14. So it’s 196. An Example of Computational Fluency
1 2 1 Problems That Require Students to Think Flexibly about Rational Numbers Using the points you are given on the number line above, locate 1/2, 2 1/2, and 1/4. Be prepared to justify your answers. 1
Resourceful Problem Solving How many different pattern block arrangements will cover a yellow hexagon?