1 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Getting Started with the Practices Investigations in Number, Data, and Space K-5 “In this changing world, those who understand and can do mathematics will have significantly enhanced opportunities and options for shaping their futures. Mathematical competence opens doors to productive futures. A lack of mathematical competence keeps those doors closed….All students should have the opportunity and the support necessary to learn significant mathematics with depth and understanding.” NCTM (2000, p.50)

2 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Getting Started with the Practices 2 Underlying Frameworks National Council of Teachers of Mathematics The 6 Principles Equity Curriculum Teaching Learning Assessment Technology NCTM (2000). Principles and Standards for School Mathematics. Reston, VA: Author.

3 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Getting Started with the Practices 3 Underlying Frameworks National Council of Teachers of Mathematics Equity “Excellence in mathematics education requires equity – high expectations and strong support for all students.” (NCTM, 2000, p.12) NCTM (2000). Principles and Standards for School Mathematics. Reston, VA: Author.

4 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Getting Started with the Practices 4 Underlying Frameworks National Council of Teachers of Mathematics Curriculum “A curriculum is more than a collection of activities: it must be coherent, focused on important mathematics, and well articulated across the grades.” (NCTM, 2000, p.14) NCTM (2000). Principles and Standards for School Mathematics. Reston, VA: Author.

5 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Getting Started with the Practices 5 Underlying Frameworks National Council of Teachers of Mathematics Teaching “Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well.” (NCTM, 2000, p.16) NCTM (2000). Principles and Standards for School Mathematics. Reston, VA: Author.

6 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Getting Started with the Practices 6 Underlying Frameworks National Council of Teachers of Mathematics Learning “Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge.” (NCTM, 2000, p.20) NCTM (2000). Principles and Standards for School Mathematics. Reston, VA: Author.

7 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Getting Started with the Practices 7 Underlying Frameworks National Council of Teachers of Mathematics Assessment “Assessment should support the learning of important mathematics and furnish useful information to both teachers and students…Assessment should not merely be done to students: rather, it should also be done for students, to guide and enhance their learning.” (NCTM, 2000, p.22) NCTM (2000). Principles and Standards for School Mathematics. Reston, VA: Author.

8 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Getting Started with the Practices 8 Underlying Frameworks National Council of Teachers of Mathematics Technology “Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students’ learning.” (NCTM, 2000, p.24) NCTM (2000). Principles and Standards for School Mathematics. Reston, VA: Author.

9 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Getting Started with the Practices Standards for Mathematical Practice “The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education.” (CCSS, 2010)

10 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Getting Started with the Practices 10 Underlying Frameworks National Council of Teachers of Mathematics 5 Process Standards Problem Solving Reasoning and Proof Communication Connections Representations NCTM (2000). Principles and Standards for School Mathematics. Reston, VA: Author.

11 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Getting Started with the Practices 11 Underlying Frameworks Strands of Mathematical Proficiency Strategic Competence Adaptive Reasoning Conceptual Understanding Productive Disposition Procedural Fluency NRC (2001). Adding It Up. Washington, D.C.: National Academies Press.

12 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Getting Started with the Practices 12 Strands of Mathematical Proficiency 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.

13 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Getting Started with the Practices 13 Standards for Mathematical Practice and PARCC 1.Make sense of problems and persevere in solving them. 2.Reason abstractly and quantitatively. 3.Construct viable arguments and critique the reasoning of others. 4.Model with mathematics. 5.Use appropriate tools strategically. 6.Attend to precision. 7.Look for and make use of structure. 8.Look for and express regularity in repeated reasoning.

14 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Getting Started with the Practices Construction of Ideas

15 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Getting Started with the Practices 15 Problem-Based or Inquiry Approach When students explore a problem and the mathematical ideas are later connected to that experience. It is through inquiry that students are activating their own knowledge and trying to make new knowledge (meaning). This builds conceptual understanding.

16 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Getting Started with the Practices 16 Procedural Fluency Knowledge and use of rules and procedures used in carrying out mathematical processes and also the symbolism used to represent mathematics. The ineffective practice of teaching procedures in the absence of conceptual understanding results in a lack of retention and increased errors.

17 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Getting Started with the Practices 17 Six Components of Mathematics Classrooms 1.Creating an environment that offers all students an equal opportunity to learn 2.Focusing on a balance of conceptual understanding and procedural fluency 3.Ensuring active student engagement in the mathematical practices 4.Using technology to enhance understanding 5.Incorporating multiple assessments aligned with instructional goals and mathematical practices 6.Helping students recognize the power of sound reasoning and mathematical integrity

18 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Getting Started with the Practices What Teachers Think What part(s) of the Investigations program has had a strong impact on student learning? Deeper thinking and understanding Students show work with pictures, numbers and words Students can now decompose numbers Mathematical reasoning has improved Hands-on learning is motivating Students are able to think flexibly Closely aligned with the common core state standards The program is completely student-centered Mental math skills have improved Less math concepts in more depth

19 National Council of Supervisors of Mathematics Illustrating the Standards for Mathematical Practice: Getting Started with the Practices Jean C. Richardson Math Specialist K-8 Mayfield City School District