Empowering Learners through the Common Core State Standards Juli K. Dixon, Ph.D. University of Central Florida juli.dixon@ucf.edu

Solve this… 3 ÷ 1/7

Solve this… 3 ÷ 1/7 Tell someone near you how you solved it.

Perspective… When asked to justify the solution to 3 ÷ 1/7 A student said this…

Perspective… When asked to justify the solution to 3 ÷ 1/7 A student said this… “Just change the division sign to multiplication and flip the fraction after the sign. 3 ÷ 1/7 becomes 3 x 7/1. So I find 3/1 x 7/1 which is 21/1 or 21.”

Perspective… When asked to justify the solution to 3 ÷ 1/7 A student said this… “Just change the division sign to multiplication and flip the fraction after the sign. 3 ÷ 1/7 becomes 3 x 7/1. So I find 3/1 x 7/1 which is 21/1 or 21.” Is this an acceptable justification?

Perspective… When asked to justify the solution to 3 ÷ 1/7 Another student said this… “I know there are 7 groups of 1/7 in one whole. Since there are three wholes, I have 3 x 7 or 21 groups of 1/7 in 3 wholes so 3 ÷ 1/7 = 21.”

Perspective… When asked to justify the solution to 3 ÷ 1/7 Another student said this… “I know there are 7 groups of 1/7 in one whole. Since there are three wholes, I have 3 x 7 or 21 groups of 1/7 in 3 wholes so 3 ÷ 1/7 = 21.” How is this justification different and what does it have to do with State Standards?

Background of the CCSSM (Common Core State Standards) Published by the National Governor’s Association and the Council of Chief State School Officers in June 2010 Result of collaboration from 48 states Provides a focused curriculum with an emphasis on teaching for depth

Background of the CCSSM 45 States + DC have adopted the Common Core State Standards Minnesota adopted the CCSS in ELA/literacy only

Background of the CCSSM “… standards must address the problem of a curriculum that is ‘a mile wide and an inch deep.’ These Standards are a substantial answer to that challenge” (CCSS, 2010, p. 3).

CCSSM Content Standards Wordle

Content Standards Critical Areas – much like our big ideas Domains – group related clusters Clusters – group related standards Standards – define what students should know and be able to do

Background of the CCSSM The CCSSM consist of Content Standards and 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” (CCSS, 2010, p. 6).

Making Sense of the Mathematical Practices NCTM Process Standards: Problem Solving Reasoning and Proof Communication Representation Connections

Making Sense of the Mathematical Practices NRC Strands of Mathematical Proficiency: Adaptive Reasoning Strategic Competence Conceptual Understanding Procedural Fluency Productive Disposition

Standards for Mathematical Practice Wordle

Making Sense of the Mathematical Practices The 8 Standards for Mathematical Practice: Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments and 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

Impact on Depth… (CCSS) Domain Number & Operations in Base Ten 4.NBT Use place value understanding and properties of operations to perform multi-digit arithmetic 5. Multiply multi-digit numbers using strategies based on place value and the properties of operations. Illustrate and explain the calculations by using equations, rectangular arrays, and/or area models. Cluster Standard

Solve this…

Solve this…

What did you do?

Perspective… What do you think fourth grade students would do? How might they solve 4 x 7 x 25?

Gr. 4 Seg 3

Perspective… Are you observing this sort of mathematics talk in classrooms? Is this sort of math talk important?

Perspective… What does this have to do with the Common Core State Standards for Mathematics (CCSSM)?

With which practices were the fourth grade students engaged? The 8 Standards for Mathematical Practice: Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments and 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

With which practices were the fourth grade students engaged? The 8 Standards for Mathematical Practice: Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments and 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

Impact on Depth… What does it mean to use strategies to multiply? When do students begin to develop these strategies?

Impact on Depth… (CCSS) Operations & Algebraic Thinking 3.OA Understand properties of multiplication and the relationship between multiplication and division. 5. Apply properties as strategies to multiply and divide… Multiply and divide within 100. 7. Fluently multiply within 100, using strategies such as the relationship between multiplication and division or properties of operations...

Impact on Depth… (CCSS) Operations & Algebraic Thinking 3.OA Understand properties of multiplication and the relationship between multiplication and division. 5. Apply properties as strategies to multiply and divide… Multiply and divide within 100. 7. Fluently multiply within 100, using strategies such as the relationship between multiplication and division or properties of operations...

What does it mean to use strategies to multiply? Consider 6 x 7

What does it mean to use strategies to multiply? Consider 6 x 7 What strategies can we use?

What does it mean to use strategies to multiply? Consider 6 x 7 What strategies can we use? How might this sort of thinking influence the order in which facts are introduced in grade 3?

Gr 3 Seg. 1

With which practice were the third grade students engaged? The 8 Standards for Mathematical Practice: Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments and 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

With which practice were the third grade students engaged? The 8 Standards for Mathematical Practice: Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments and 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

How do we support this empowerment? “… a lack of understanding [of mathematical content] effectively prevents a student from engaging in the mathematical practices” (CCSS, 2010, p. 8). When and how do we develop this understanding?

Where do we start? ….in Kindergarten.

Making Sense of Numbers to 20 How do we help students to truly make sense of number patterns?

Making Sense of Numbers to 20 How do we help students to truly make sense of number patterns? Consider number pairs for 8.

Describing the Standards K.OA.3: Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).

Describing the Standards Watch the following video of a Kindergarten class as they explore Number Pairs for 8.

Describing the Standards Watch the following video of a Kindergarten class as they explore Number Pairs for 8. Look for how patterns are addressed.

Gr K. Seg 2

Which practice was of focus here? The 8 Standards for Mathematical Practice: Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments and 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

Which practice was of focus here? The 8 Standards for Mathematical Practice: Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments and 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

Engaging Students in Reasoning and Sense Making We need to question students when they are wrong and when they are right. We need to create an environment where students are expected to share their thinking. We need to look for opportunities for students to reason about and make sense of mathematics.

What is your biggest take away regarding the practices and your role? The 8 Standards for Mathematical Practice: Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments and 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

How do we support this empowerment? Teachers need content knowledge for teaching mathematics to know the tasks to provide, the questions to ask, and how to assess for understanding. Math Talk needs to be supported in the classroom. Social norms need to be established in classroom and professional development settings to address misconceptions in respectful ways.

How do we support this empowerment? What we know best might be the most difficult to change. We need to be aware of the changes so that we can prepare ourselves to meet the needs of our students.

Empowering Learners through the Common Core State Standards Juli K. Dixon, Ph.D. University of Central Florida juli.dixon@ucf.edu