Wisconsin’s Common Core Math Standards

Meaningful Mathematics Through Building Mathematical Practices

Hank Kepner, National Council of Teachers of Mathematics, Past-President

Univ. of Wisconsin-Milwaukee

Milwaukee Public Schools

kepner@uwm.edu

Common Core State Standards

Following over 2 decades of math standards development and refinement initiated by National Council of Teachers of Mathematics:

•Spring,2009. National Governors’ Association (NGA) & the Council of Chief State School Officers (CCSSO) agreed to support a common core of state standards, starting in Mathematics and Language Arts.

•Fall, 2009. College- and Career Readiness Math Standards,

•May, 2010. Common Core State Standards in Mathematics and English/Language Arts released for approval by the individual states.

•June 2, 2010. Wisconsin adopted the Math, English/Language Arts standards.

Common Core Standards

• Standards for Mathematical Practice (pp 6-8)

• K – 8 Grade level standards

• High School standards–“conceptual categories”

Standards for Mathematical Practice

• Mathematically proficient students:– Make sense of problems and persevere in

solving them– Reason abstractly & quantitatively– Construct viable arguments and critique the

reasoning of othersStudent behaviors!

Standards for Mathematical Practice

• Mathematically proficient students:– Model with mathematics – Use appropriate tools strategically– Attend to precision– Look for and make use of structure– Look for & express regularity in repeated

reasoning» Student behaviors!

NCTM – Principles & Standards for School Mathematics

• Process Standards– Problem Solving– Reasoning and Proof– Communication– Connections– Representation

• PRINCIPLE– Technology

Mathematical ProficiencyNRC Adding It Up (2001)

• conceptual understanding – comprehension of

mathematical concepts, operations, and relations

• procedural fluency – skill in carrying out procedures flexibly, accurately, efficiently, and appropriately

• strategic competence – ability to formulate, represent, and solve mathematical problems

• adaptive reasoning – capacity for logical thought, reflection, explanation, and justification

• productive disposition – habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.

Construct Viable Arguments &Critique the Reasoning of Others

• Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments.

• They make conjectures and build a logical

progression of statements to explore the truth of their conjectures.

Construct Viable Arguments &Critique the Reasoning of Others

• They are able to analyze situations by

breaking them into cases, and can recognize and use counterexamples.

• They justify their conclusions, communicate them to others, and respond to the arguments of others.

• They reason inductively about data, making plausible arguments that take into account the context from which the data arose.

Construct Viable Arguments &Critique the Reasoning of Others

• Mathematically proficient students are able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is.

Construct Viable Arguments &Critique the Reasoning of Others

• Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies.

• Students at all grades can listen to or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Mathematics - Literacy

• Determine the meaning of symbols, key terms, & math-specific words. – Long noun phrases: “a prism with sides 8,10,12cm”– Verbs constructing relational processes:

• Attributive – constructs information about membership• Identifying – constructs relations of identity and equality

– Conjunctions: if, when, therefore, and, or….– Translations between mathematical expressions

and common language.• Compare & contrast information obtained..

Mathematics - Literacy

• More than any other discipline, the construction of knowledge about mathematics depends on the oral language explanations and interactions of the teacher.

• Students need to participate orally and in writing by explaining solution processes, describing conjectures, proving conclusions, & presenting arguments.

Content Standards K-12

A transition – NOT a rushed jump - from the – MPS Learning Targets and– Wisconsin Model Academic Standards

To the Wisconsin Core Standards

Content Standards K- 8 Structure

•Domain

–Cluster

•Standards

Document Structure: Grade 4 Domains/Clusters

• Operations and Algebraic Thinking• Use the four operations with whole numbers to solve problems

• Gain familiarity with factors and multiples

• Generate and analyze patterns

• Number & Operations in Base Ten• Generalize place value understanding for multi-digit whole numbers

• Use place value understanding and properties of operations to perform multi-digit arithmetic

• Number & Operations: Fractions• Extend understanding of fraction equivalence and ordering

• Build fractions from unit fractions…from whole number operations.

• Understand decimal notation for fractions, compare.

Document Structure: Grade 4 Domains/Clusters

• Measurement and Data• Solve problems involving measurement &

conversion from a larger unit to a smaller unit.

• Represent and interpret data.• Geometric measurement: understand concepts

of angle and measure angles.

• Geometry• Draw and identify lines and angles; and classify

shapes by properties of their lines and angles.

Document Structure: Grade 4 Geometry Cluster/Standards

• Draw and identify lines & angles; and classify shapes by properties of their lines and angles.1. Draw points, lines, line segments, rays, angles (right, acute,

obtuse), & perpendicular and parallel lines. Identify these in 2-dimensional figures.

2. Classify 2-d figures based on the presence or absence of parallel or perpendicular line, or the presence or absence of angles of specified sizes. Recognize right triangles as a category, and identify right triangles.

3. Recognize lines of symmetry for a 2-d figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify or draw lines of symmetry

K-5 Domains

Counting & Cardinality K

Operations, Algebraic Thinking 1-5

Number & Operations in Base 10 1-5

Number & Operations–Fractions1-5

Measurement and Data 1-5

Geometry 1-5

Place Value, Base 10 Trajectory

• K– Foundation of bundling– Emphasis on teens

• Grade 1– Extends to 10, 20, 30,…– Learn to compare using place value

• Grade 2– Extend to 100 as bundle of ten 10’s– Extend to 100, 200, 300,…– Expand notation and comparison

• Grade 3– Use place value understanding, properties of

operations to perform multi-digit arithmetic. (Add, subtract to 1000, multiply multiples of 10 by one-digit numbers)

Place Value, Base10 Trajectory

• Grade 4– Use place value understanding, properties of operations to

perform multi-digit arithmetic. (Generalize:each digit is ten times the next; compare large numbers; Multiply 2-digit by 2-digit and 1-digit by 4-digit numbers.)

– Understand decimal notation for fractions and compare decimal fractions.

• Grade 5– Understand the place value system.

– Understand decimal notation for fractions and compare decimal fractions.

• Grade 8– Work with radicals and integer exponents.

Value of Learning Trajectories for Teachers

• Know what to expect about students’ preparation.

• Know what teachers in the next grade expects of your students.

• Identify clusters of related concepts at grade.

• Clarity about student thinking, discourse• Engage in rich uses of classroom

assessment

Content Focus on Conceptual Understanding

• Avoids too much emphasis on procedure• Facilitates student flexibility• Stronger focus on multiple representations• Application to practical situations• Accurate, clear explanations• Justified conclusions• Strategic use of technology

Middle Grades Domains

Ratio-Proportional Relationships6-7

The Number System 6-8

Expressions & Equations 6-8

Functions 8*

Geometry 6-8

Statistics & Probability 6-8

High School Categories/Clusters

• Number and Quantity– The Real Number System– Quantities– The Complex Number System– Vector and Matrix Operations

• Algebra– Seeing structure in expressions– Arithmetic with Polynomials, Rational Expressions– Creating Equations– Reasoning with Equations and Inequalities

• Functions – Interpreting functions– Building functions– Linear, quadratic and exponential models– Trigonometric Functions

High School Categories/Clusters

• Modeling

• Geometry– Congruence

– Similarity, Right Triangles and Trigonometry

– Circles

– Expressing Geometric Properties with Equations

– Geometric Measurement and Dimension

– Modeling with Geometry

• Statistics and Probability– Interpreting categorical & quantitative data

– Making Inferences & Justifying Conclusions

– Conditional Probability and Rules of Prob.

– Using Probability to Make Decisions

Strengths of the Wisconsin Core

• Focus attention on core concepts in number & numeration and their relationships to operations – structure of the number system – as it expands.

• Tie decimal reasoning strongly to fractions.

• More aggressive timelines for teaching particular concepts in elementary & middle grades – geometry, basic algebra.

• Support articulation of key learning trajectories in numeration and geometry.

Next Steps for the Wisconsin Core

• The Mathematics Practice Standards are presented independent of content standards – risk of being isolated or ignored.

• There is limited attention to engaging students in mathematics through modeling with a robust and intriguing use of technology and skills required.

• The wording of many standards is obtuse, combining math propositions with combination of content & pedagogical advice, verbs hard to interpret for assessment.

Lack of Engagement in Math

• Lack of student engagement in learning is the greatest problem in many math classrooms.

• The Mathematics Practice Standards present a way to leverage discourse.

• Formative assessment or Assessment for Learning as learning activity for teachers and students.

Effective Formative Assessment Strategies

• Clarifying learning intentions and sharing criteria for success

• Engineering effective classroom discourse• Providing feedback - moves learners forward• Activating students as owners of their

own learning• Activating students as instructional

resources for one another

Re-teaching vs. Re-engagement

Re-teaching Re-engagementTeach unit again Revisit student thinking

Address missing basic skills Address conceptual understanding

On the same or similar problems

Examine task from different perspective

Practice more Critique S’s approaches

Cognition lower Cognition higher