mathematics organizations national council of teachers of mathematics and collaborators hank kepner...
TRANSCRIPT
Mathematics Organizations
National Council of Teachers of Mathematics
and Collaborators
Hank Kepner
National Council of Teachers of Mathematics, Past-President
University of Wisconsin-MilwaukeeMilwaukee Public Schools
Disclaimer
• The instructional practices and assessments discussed or shown are not an endorsement by the U.S. Department of Education.
On-going Collaborations
Advisory Group for CCSS Mathematics• NCTM• Natl Council of Supervisors of Mathematics (NCSM)• Assoc. of Mathematics Teacher Educators (AMTE)• Assoc. of State Supervisors of Mathematics (ASSM)• Council of Chief State School Officers (CSSO)• National Governors Association (NGA)• SMARTER Balanced Assessment Consortium• Partnership for the Assessment of Readiness for
College and Careers (PARRC, Achieve)
On-going CollaborationsMathematical Assoc. of American and NCTM (Joint Committee on Mutual Concerns)
• A long history on the study of Calculus• Current focus on student transition from high school
to college – addressing• Examination of Quantitative Literacy as goal for all
American Statisical Association & NCTM (Joint Committee)
• A long history of establishing quantitative literacy and materials for use in secondary schools.
• Current focus on Common Core expectations for statistics and probability –their learning progressions, curriculum materials, and professional development
On-going Collaborations
American Mathematical Association of Two-Year Colleges
Carnegie Foundation Support• Alternative mathematics pathways for non-STEM college
students administered by the Carnegie Foundation for the Advancement of Teaching.
• Statway - a year-long sequence that takes students to and through college-level statistics.
• Quantway - a one-semester quantitative reasoning coursethat incorporates fundamental numeracy & algebraic literacy.
• Courses developed by members of leading professional organizations including NCTM, MAA, AMS, ASA, and others.
On-going Collaborations
Response to Intervention – A National Agenda
NCTM & Council for Exceptional Children
NSF-supported conference building:* A research agenda and
* Professional development programsfor instructional interventions in
mathematics
History Informing My Remarks
• 1980 An Agenda for Action
• 1989 Curriculum and Evaluation Standards for School Mathematics– Addenda series
• 1991 Professional Teaching Standards
• 1993 Assessment Standards
History Informing My Remarks
• 2000 Principles and Standards for School Mathematics– Navigations series
• 2006 Curriculum Focal Points – Grade Level/ Band books
• 2009 Focus in High School Mathematics: Reasoning and Sense Making
• Statistics/Probability, Algebra, Geometry, Equity, Technology
Common Core State StandardsFollowing 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 develop a common core of state standards, starting in Mathematics and Language Arts.
•Fall 2009. College- and Career Readiness Math Standards,
•June 2, 2010. Common Core State Standards in Mathematics and English/Language Arts
released.
Standards for Mathematical Practice
Mathematically proficient students:
• Make sense of problems and persevere in solving them
• Reason abstractly and quantitatively
• Construct viable arguments and critique the reasoning of others• These are measures of Student behavior!
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 and express regularity in
repeated reasoning • These are measures of Student behavior!
NCTM Process Standards CCSS Mathematical Practices
Problem Solving •Make sense of problems and persevere in solving them.•Use appropriate tools strategically
Reasoning and Proof •Reason abstractly and quantitatively.•Critique the reasoning of others.•Look for and express regularity in repeated reasoning•Construct viable arguments
Communication •Construct viable arguments
Connections •Attend to precision.•Look for and make use of structure
Representations •Model with mathematics.
NCTM Process Standards and the CCSS Mathematical Practices
Mathematical Proficiency Adding It Up (NRC, 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
• understand & use stated assumptions, definitions, and previously established results in constructing arguments
• make conjectures and build a logical progression of statements to explore the truth of their conjectures
• justify their conclusions; communicate them to others
• listen to or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
Equity: Mathematical reasoning and sense making must be evident in the mathematical experiences of all students.
• Courses students take have an impact on the opportunities that they have for reasoning and sense making
• Students’ demographics too often predict those opportunities
• Expectations, beliefs, and biases have an impact on the mathematical learning opportunities provided for students
Common Core - Clusters
• May appear in multiple grade levels with increasing developmental standards as the grade levels progress
• Indicate WHAT students should know and be able to do at each grade level
• Reflect both mathematical understandings and skills, which are equally important
17
Curriculum Focal Points PK-8
• NCTM example to streamline, focus and align the PreK-8 curriculum to achieve coherence.
• A resource for– CCSSMCCSSM
18
Teaching with Curriculum Focal Points
Grade Bands• Focus in Grades PK-2 • Focus in Grades 3-5• Focus in Grades 6-8
Grade Level Books• Individual Grades PK-8
Developing Essential Understanding
• Essential knowledge for teachers
• The big ideas for each book’s topic
• The mathematics that is focused, connected and useful to teachers
Developing Essential Understanding
• PK-2 Number & Numeration
• PK-2 Addition & Subtraction
• 3-5 Rational Numbers
• 5-8 Ratios, Proportions, & Proportional
Reasoning
• 9-12 Function
Forthcoming: Developing Essential Understanding
• PK-8 Reasoning and Sense Making
• PK-2 Geometry
• 3-5 Multiplication and Division
• 3-5 Algebraic Ideas & Readiness
• 3-5 Geometric Shapes and Solids
• 6-8 Expressions and Equations
• 6–8 Measurement
• 6-8 Data Analysis and Statistics
• 9-12 Geometric Relationships
• 9-12 Reasoning and Proof
• 9-12 Statistics
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 - Challenge: integrate throughout students’ experiences, explicit expectations.
• 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
Geometry
Statistics and Probability
• Analyze single and two variable data
• Understand the role of randomization in experiments (understand variation)
• Make decisions, use inference, and justify conclusions from statistical studies
• Use the rules of probability
Modeling
Modeling has no specific domains, clusters or standards. Modeling is included in the other conceptual categories and marked with a asterisk.
Focus in High School Mathematics: Reasoning and Sense Making
• Giving direction and focusto high school mathematics
• Making reasoning andsense making the core
• Making connections
www.nctm.org/hsfocus
Focus in High School Mathematics: Reasoning and Sense Making
• Topic Books
• Statistics and Probability• Algebra • Geometry• Reasoning & Sense Making for All Students• Technology (Fall,2011)
High School Course PathwaysCommon Core Appendix
• Appendix A Common Core State Standards• These are NOT mandated standards!
• Provides four possible approaches to organizing the content across several courses - by no means an exhaustive list.
• States, curriculum developers, schools may use as guides in preparing instructional materials and assessments (End of course?)
Some Strengths of the Common Core Standards
• Focus attention on core concepts in number and place value and relationships of operations prior to algorithms — structure of the number system—as it expands
• Tie decimal reasoning strongly to fractions• More aggressive timelines for teaching
particular concepts in elementary and middle grades—geometry, basic algebra
• Support articulation of key learning trajectories in number and geometry
Next Steps for the Common Core Standards
• The Mathematical Practice Standards are presented independent of content standards—risk of being isolated or ignored
• Use of and competency with technology is not adequately or constructively addressed for students
• There is limited attention to engaging students in mathematics through modeling with a robust and intriguing use of technology and skills required
• Yet to be done: Math connections—across strands and applications
• The wording of many standards is obtuse—combining math propositions with combination of content and pedagogical advice, verbs hard to interpret for assessment
Next Steps for the Common Core Standards
• The high school standards are of widely varying grain size—there is substantive need to unpack many of the standards to clarify how the sub-constructs develop and build on each other
• The document is weak in relation to “career” in the promise of “College and Career Ready Standards” • I would argue the developers gave up on this!
• The standards ignore empirical data on children’s ability to develop understanding of variability, distributions, data use, statistical reasoning and probability in early grades. Added abruptly in 6th, 7th grade—an already difficult transition period
Next Steps for the Common Core Standards
• The standards will make sense only when we have instructional and assessment exemplars to use and analyze—the operational definitions!
• Most standards do not describe depth of cognitive demand to be assessed. Caution about trivial level
• Monitor the assessment developments to ensure sound assessment – more than multiple choice:
• Department of Education: Funded Assessment Consortia to develop assessment systems for use by 2014-2015
• Partnership for the Assessment of Readiness for College and Careers (PARCC includes Illinois, Achieve)
• SMARTER Balanced Assessment Consortium (includes Wisconsin)
Key Recommendations• Work on instructional strategies and
tasks to refine your engagement of students in the Standards of Mathematical Practice.
• With colleagues, study the content in a domain and cluster chunks, often across grade levels, attending to connections –
not as isolated standards!
Implementation support from NCTM
Making It Happen:
A Guide to Interpreting and Implementing the Common Core State Standards for Mathematics
Making it Happen will include:
• How the content of CCSSM fits with NCTM’s Standards documents—including a road map to NCTM’s resources
• A deep analysis of the connections between NCTM’s Process Standards and the CCSS Mathematical Practices
• Examples of using NCTM resources to interpret and implement CCSS
Examples of Implementation support underway from NCTM
• Development of resource toolkit linked to a mobile or e-conference package
• Areas of focus on Professional Development with respect to both content and mathematical processes
• Grade band specific power points available on the NCTM website
Products of Implementation from NCTM
July 28-30 Orlando, Florida
Infusing the Classroom with Reasoning and Sense Making
An NCTM Interactive Institute on High School Mathematics. • Now more than ever students need the right tools to succeed; still, many
students lack the crucial reasoning and sense making skills necessary to apply mathematics to other contexts. NCTM’s Interactive Institute on High School Mathematics, Infusing the Classroom with Reasoning and Sense Making, is a new event designed to provide strategies for creating a high school math classroom in which students are engaged in reasoning and sense making.
www.nctm.org/reasoning
•
Products of Implementation from NCTM
August 9-11, Baltimore, MD AND throughout the Year:
Algebra Readiness for Every StudentAn NCTM Interactive Institute for Grades 3-8 with Extended Online Professional Development
Experience professional development over the whole school year by participating in NCTM’s new event dedicated to algebra readiness for grades 3-8. Kick off this Institute in Baltimore where you’ll participate in activities and network with peers from across the country, then reinforce what you learn by participating in online interactive workshops and discussion groups over the school year.
• • Orlando, Florida—July 28-30, 2011 • Now more than ever students need the right tools to succeed; still, many students lack the crucial reasoning and sense making skills necessary to apply mathematics to other contexts. NCTM’s Interactive Institute on High School Mathematics, Infusing the Classroom with
Reasoning and Sense Making, is a new event designed to provide strategies for creating a high school math classroom in which students are engaged in reasoning and sense making Space is limited so mark your calendars now for July 28-30, 2011. Registration opens in January, but in the meantime, learn everything you need to know to start planning at www.nctm.org/reasoning.
•
•
•
• Orlando, Florida—July 28-30, 2011 • Now more than ever students need the right tools to succeed; still, many students lack the crucial reasoning and sense making skills necessary to apply mathematics to other contexts. NCTM’s Interactive Institute on High School Mathematics, Infusing the Classroom with
Reasoning and Sense Making, is a new event designed to provide strategies for creating a high school math classroom in which students are engaged in reasoning and sense making Space is limited so mark your calendars now for July 28-30, 2011. Registration opens in January, but in the meantime, learn everything you need to know to start planning at www.nctm.org/reasoning.
•
Speak at a Conference• Speak at an NCTM Conference • Advertise, Exhibit or Sponsor• Annual Meeting Sponsorships Advertise With NCTM •
NCTM Conference Blog• The Fun Starts on Monday!
Wed, 09 Mar 2011 Get Ready, Get Set...for Indianapolis!Thu, 24 Feb 2011 Technology Themed Sessions are now postedTue, 15 Feb 2011
• RSS Feed •
•
Learning Trajectories in Mathematics – Our Next Step
The Common Core initiative was to be built on, what turned out to be extremely limited, knowledge about student learning trajectories and associated teaching trajectories.
This effort is in its infancy. However, the focus is critical to building learning environments/tasks where students are supported in making sense of the mathematical knowledge they are expected to have and to use – in math – but more importantly in an unbelievable number of settings throughout their life and employment.
CPRE (2011) Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction. http://www.cpre.org/ccii/
