WELCOME Your Faculty Kevin Baird Kevin.Baird@CommonCoreInstitute.Org
Today’s Webinar Lecture Eight Standards for Mathematical Practices In Depth
Today’s Overview Implications from ELA: CCR Number One Knowledge: Math Practices Review Grouping the Standards for Math Practices Learning Progressions Research Basis Resources: Deconstructed Standards Math Fluency Standards Handouts
Scope & Sequence Readings: A few more introduced today Spend time with the Progressions Spend time with the actual CCSS (Math is more focused) NOTES Today: “Big Rocks”
AN Eye on Skill Building:What Teachers CAN DO.. Safety in Asking for Help A Little Review
The National Pathway A Process of Discovery, Support and Mastery
The Alignment Process Establish Goals Measure to Focus on Priorities Align Practice to Goals & Priorities Curriculum Instruction Assessment Observe, Communicate, Teach, Direct Refinement
Mathematics More than Reading, Mathematics is often Curriculum Driven Curriculum Materials are unlikely to help (yet) or make up for lack of math skills
Plan Plan Plan Plan Plan Plan: Provides the direction to eliminate the treadmill effect. Vision: The “Why are we doing this?” to combat confusion. Skills: The skill sets needed to combat anxiety. Incentives: Reasons, perks, advantages to combat resistance Resources: Tools and time needed to combat frustration. Conditions for Successful Implementation Knoster, T., Villa, R., & Thousand, J. (2000)
Key Data Review • College & Career Readiness Benchmarks • The Assessment Today • Middle Grades
College Readiness Benchmarks by Subject Percent of ACT-Tested School Graduates Meeting College Readiness Benchmarks By Subject 2011 66% of all ACT-tested high school graduates met the English College Readiness Benchmark in 2011. Just 1 in 4 (25%) met all four College Readiness Benchmarks. In 2011, 52% of graduates met The Reading Benchmark, while 45% met the Mathematics Benchmark. Just under 1 in 3 (30%) Met the College Readiness Benchmark in Science.
Structure of Standards • Focus • Coherence • Progression • Fluency • Language
Third First Second Kindergarten Number Sense Operations Measurement Consumer Applications Basic Algebra Advanced Algebra Geometric Concepts Advanced Geometry Data Displays Statistics Probability Analysis Trigonometry Special Topics Functions Instructional Technology • Memorize Facts, Definitions, Formulas • Perform Procedures • Demonstrate Understanding • Conjecture, Analyze, Generalize, Prove • Solve Non-Routine Problems/Make Connections
Learning Progression Standard: Identify the relative position of simple positive fractions, positive mixed numbers, and positive decimals and be able to evaluate the values based on their position on a number line. Compare fractions, decimals and mixed numbers by identifying their relative position on a number line Identify and locate the approximate location of decimals in hundredths on a number line Indicate the approximate location of thirds, fourths, and fifths on a number line Locate tenths in decimal form on a number line Place halves in fraction form on a number line Locate simple whole numbers on a number line Draw a basic number line from 0 to 10
PROFICIENCY Grade 8: Critical Areas In Grade 8, instructional time should focus on three critical areas: (1) formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations; (2) grasping the concept of a function and using functions to describe quantitative relationships; (3) analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and understanding and applying the Pythagorean Theorem. FLUENCY
Kindergarten Essential vocabulary ELA Math Attribute Decompose Decomposition Composition Hexagon Dimensional Vertices Category • Stanza • Preference • Punctuation • Collaborate • Illustrator • Brainstorm • Punctuation • Non-fiction
Standards for Mathematical Practice Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments / critique the reasoning of others Model with mathematics Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning
IN A NUTSHELL The Process of “Figuring Out” and “Problem Solving” is at least as important as the solution itself “There is no ‘right’ answer… some answers are more correct than others”
Background Influence Understanding The Early Findings
2008 National Math Advisory Panel Key Findings • Fall off begins late middle school, where Algebra starts • The sequence of learning matters • Curriculum should be streamlined and focused • Combine conceptual understanding, procedural fluency, and automatic (quick, effortless) recall of facts • Effort, not just talent, counts
Benchmarks • By the end of the third grade, students should be proficient in adding and subtracting whole numbers. • Two years later, they should be proficient in multiplying and dividing them. • By the end of the sixth grade, the students should have mastered the multiplication and division of fractions and decimals.
To prepare students for algebra, the curriculum must simultaneously develop conceptual understanding, computational fluency and problem-solving skills. • Findings closely tracked the 2006 NCTM Report
The Problem with Fractions • Fractions are especially troublesome for Americans, the report found. It pointed to the National Assessment of Educational Progress, standardized exams known as the nation’s report card, which found that almost half the eighth graders tested could not solve a word problem that required dividing fractions. Panel members said the failure to master fractions was for American students the greatest obstacle to learning algebra. Just as “plastics” was the catchword in the 1967 movie “The Graduate,” the catchword for math teachers today should be “fractions,” said Francis Fennell, president of the National Council of Teachers of Mathematics.
2011 NAEP MAKING PROGRESS LOOK AT HIGHLY QUALIFIED!
Common core: Higher Rigor • By Grade 4: BOTH: fluent at adding, subtracting, and multiplying whole numbers; apply place value; classify two-dimensional geometric figures. COMMON CORE: Multiply fractions by whole numbers, understand multiplication algorithms through applications of place value and some operations.
By 8th: BOTH: Focus on algebraic expressions, equations, and functions; expect effective work with symbols to transform linear expressions and solve linear equations (beginning algebra). COMMON CORE: Emphasis on geometric properties, using function to model relationships between quantities.
High School BOTH: Very Similar Content, Different Sequence & Combination COMMON CORE: Expects more rigor with polynomial functions and attributes
Brown & Quinn 2006 1. Children in the early primary grades should be allowed the time to develop whole number concepts and whole number operations informally with abundant concrete referents. Arabic symbols should be used for counting purposes only and always connected to concrete objects or pictorial representations. Informal practice with fraction concepts should be limited to experiences that arise naturally, like fair sharing or situations that involve money. Lamon (1999) claims that studies have shown that if children are given the time to develop their own reasoning for at least three years without being taught standard algorithms for operations with fractions and ratios, then a dramatic increase in their reasoning abilities occurred, including their proportional thinking (p. 5).
Brown & Quinn 2006 2. Upper primary students should be given experiences that extend the whole number concept with an eye toward algebra involving an informal treatment of the field properties. These students need to be provided with experience in partitioning as a method for solving verbal problems involving fractions (Lamon, 1999; Huinker, 1998). The informal treatment of fractions should include manipulation of concrete objects and the use of pictorial representations, such as unit rectangles and number lines. Fraction notation must be developed, but formal fraction operations using teacher-taught algorithms should be postponed. Learning the subject of fractions will revolve around informal strategies for solving problems involving fractions. The objective at this level is to build a broad base of experience that will be the foundation for a progressively more formal approach to learning fractions.
3. In middle school, the development of fraction operations as an extension of whole number operations should provide experiences that guide and encourage students to construct their own algorithms (Lappan & Bouck, 1998; Sharp, 1998). More time is needed to allow students to invent their own ways to operate on fractions rather than memorizing a procedure (Huinker, 1998). Progressively this development should lead to more formal definitions of fraction operations and algorithms that prepare students for the abstractions that arise later in the study of algebra (Wu, 2001). How fractions should be taught is inexorably linked to when the concepts are being presented and what impact the learned concepts will have on future mathematics courses
Rigor “Students who complete Algebra II are more than twice as likely to graduate from college compared to students with less mathematical preparation.”
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