State of the State Mathematics K-12

What’s New? Next Generation Content Standards and Objectives and Standards for Mathematical Practice Smarter Balanced Assessment Elementary Mathematics Specialist Math I Certification High School Math Course Sequence Math I Lab

Key Advances in Mathematics 1.Standards for Mathematical Practice 2.Properties of operations: Their role in arithmetic and algebra 3.Mental math and “algebra” vs. algorithms 4.Operations and the problems they solve 5.Units and unitizing a.Unit fractions b.Unit rates 6.Defining congruence and similarity in terms of transformations. 7.Quantities-variables-functions-modeling 8.Number-expression-equation-function 9.Modeling

Suggested First Implementation Steps: Mathematical practices Progressions within and among content clusters and domains Key advances Local assessments –Classroom formative and summative assessment –State released tasks

Reaching the Goal Report – EPIC 2011

Next Generation Standards Progression K HS Counting & Cardinality Number and Operations in Base Ten Ratios and Proportional Relationships Number & Quantity Number and Operations – Fractions The Number System Operations and Algebraic Thinking Expressions and EquationsAlgebra Functions Geometry Measurement and DataStatistics and Probability Statistics & Probability

Grade Priorities in Support of Rich Instruction and Expectations of Fluency and Conceptual Understanding K–2 Addition and subtraction, measurement using whole number quantities 3–5 Multiplication and division of whole numbers and fractions 6 Ratios and proportional reasoning; early expressions and equations 7 Ratios and proportional reasoning; arithmetic of rational numbers 8Linear algebra Priorities in Mathematics

GradeRequired Fluency KAdd/subtract within 5 1Add/subtract within 10 2 Add/subtract within 20 Add/subtract within 100 (pencil and paper) 3 Multiply/divide within 100 Add/subtract within Add/subtract within 1,000,000 5Multi-digit multiplication 6 Multi-digit division Multi-digit decimal operations 7Solve px + q = r, p(x + q) = r 8Solve simple 2  2 systems by inspection Key Fluencies

Course Descriptions Math I – 6 units Creating equations Function families – linear and exponential Systems Descriptive Statistics Congruence, Proof and Constructions Connecting Algebra and Geometry through Coordinates

Math II – 6 units Extending the number system (includes polynomials and complex numbers) Quadratic functions and modeling Expressions and equations Application of Probability Similarity, Right Triangle Trigonometry, and Proof Circles With and Without Coordinates

4 th course options document

Learning Progressions

A powerful organizing principle of the Next Generation West Virginia State Standards is that of learning progressions, where an idea is reinforced over grade levels to build depth of understanding.

Standards for Mathematical Content: Learning Progressions “the development of these Standards began with research-based learning progressions detailing what is known today about how students’ mathematical knowledge, skill, and understanding develop over time.” -CCSS Mathematics, p. 4

WV’s high school pathway allows for progressions to be developed that relate to important ideas in high school mathematics.

For example: Linear, Quadratic and Exponential Models Math I linear and exponential functions, Math II extending the ideas to quadratic functions Math III incorporating logarithms.

In contrast, the alternate track squeezes quadratic functions into Algebra I and has a year hiatus from algebraic modeling during the Geometry course.

Grade 8 M.8.F.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.

Math I M.1HS.LER.7 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. (Focus on linear and exponential functions.)

Math II M.2HS.QFM.1 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

Pythagorean Theorem: Grade 8 M.8.G.6 Explain a proof of the Pythagorean Theorem and its converse. M.8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. M.8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Pythagorean Theorem: Math I M.1HS.CAG.3 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. (Provides practice with the distance formula and its connection with the Pythagorean theorem.)

Pythagorean Theorem: Math II M.2HS.SPT.13 Prove the Pythagorean identity sin 2 (θ) + cos 2 (θ) = 1 and use it to find sin (θ), cos (θ), or tan (θ), given sin (θ), cos (θ), or tan (θ), and the quadrant of the angle. In this course, limit θ to angles between 0 and 90 degrees. Connect with the Pythagorean theorem and the distance formula. Extension of trigonometric functions to other angles through the unit circle is included in Mathematics III.

Message from NCTM President by NCTM President J. Michael Shaughnessy NCTM Summing Up, March 2011 In my view, the “layer cake” approach to high school mathematics that currently dominates so many secondary school mathematics programs—built on course sequences such as Algebra I, Geometry, Algebra II, or Algebra I, Algebra II, Geometry—is an outmoded approach in a 21st-century educational system.

Special High School Challenges Deeply entrenched practices tied to particular course names New content and perspectives Beliefs in teachers’ content expertise –Hard to get the conversations started

Excellent Math Classroom Describe what you see, hear and feel in an excellent math classroom. Prepare to share. You have five minutes.

Math Practices Rubric Student Look-fors

Video clips

Mathematical Literacy

Smarter Balanced Assessment Consortium – Where we are now… Content Specifications Item Specifications Test Design Specifications Implications

Resources AMTE, ASSM, NCSM, NCTM McCallum standards progressions

What matters are the interactions, in classrooms, among the teacher, the students and the mathematical ideas – Cohen and Ball

Q&A Documents