Insights About the Grade 1Learning Expectations in the CCSS

Topic #1: The meaning of the equal sign

Presented by the Office of Curriculum, Instruction and Student Support

Dewey Gottlieb Stacie Kaichi-ImamuraEducational Specialist for Mathematics Mathematics Resource Teacher

A Shift in Perspective

The CCSS for Mathematics compel a change in the culture of traditional mathematics classroom.

YOU are the most important variable in the equation

Your knowledge, experience, insights, creativity, energy and

willingness to step out of your comfort zone

are desperately needed to improve K-12 mathematics education.

Instruction with Intention

Effective teachers

• plan instruction and design learning opportunities with the understanding that at its core, learning is about transforming information into knowledge.

Instruction that Makes a Difference

Effective teachers

• design learning opportunities with the understanding that at the heart of effective mathematics instruction is an emphasis on problem solving, reasoning and sense-making.

What we hope you’ll walk away with

Our Intentions:

• Enhance teachers’ content knowledge for teaching mathematics.

• Provide insight about what a particular Common Core standard (or cluster of standards) really means.

Our Intentions:

• Share ideas for engaging students in learning opportunities aimed at promoting deeper understanding of important mathematical ideas.

What we hope you’ll walk away with

Teaching to the Big Ideas in the CCSS

Domain: Operations and Algebraic Thinking

Clusters:• Represent and solve problems involving

addition and subtraction.• Add and subtract within 20.• Work with addition and subtraction

equations.

What do we really want students to know?

1.OA.7: Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false.

What does “=“ mean?

What does “=“ mean?

The equal sign? Equals means

equals.

The equal sign? Equals means

equals.

Teaching to the Big Ideas in the CCSS

Domain: Operations and Algebraic Thinking

Clusters:• Represent and solve problems involving

addition and subtraction.• Add and subtract within 20.• Work with addition and subtraction

equations.

Why should I reconsider how I teach “equality”?

A common misconception:

3 + 4 = + 5

2 + = 3 + 4

3 + 4 = 2 +

Why should I reconsider how I teach “equality”?

Setting the foundation for …

1/3 = 3/9

3/4 = 75%

(3 + 4) + 5 = 3 + (4 + 5)

What does the equal sign really mean?

=•It states a relationship between two quantities

• If two quantities are equal, they represent the same amount

• EQUALS means “THE SAME AS”

What do we really want students to know?

Progressing through concrete, pictorial and abstract representations of mathematical ideas

Instruction with Intention

Working with actual objects: concrete

Progressing through concrete, pictorial and abstract representations of mathematical ideas

Instruction with Intention

=

Creating and manipulating drawings: pictorial

The actions we perform on the concrete and pictorial representations should have a direct connection to the abstract representation that we want students to understand and develop expertise with.

Instruction with Intention

=

Representations with numerals and symbols: abstract

4 = 1 + 3

Students should have routine practice reading number sentences aloud (in both directions) using the phrase “is the same as”.

6 = 6

8 = 5 + 3

7 + 3 = 4 + 6

Instruction with Intention

Build connections with pictorial representations.

=

=

=

In the lesson, notice how the teacher

•activates prior knowledge

•models mathematical ideas with concrete experiences

•asks questions and uses a “think-pair-share” strategy to make all students accountable for responding to her questions.

Purposeful Instructional Design: Part I -- Engage

In the lesson, notice how the teacher•provides students opportunities to engage with the mathematics (individually and in groups)•purposefully sequences the problems

Purposeful Instructional Design: Part II -- Explore

In the lesson, notice how the teacher

•brings the class back together to wrap up the discussion, checks for understanding and answers questions.

Purposeful Instructional Design: Part III -- Summarize

In the lesson, notice how the teacher

•provides an opportunity for students to take what they have learned and apply it in a different context, moving them toward the learning goal that is expected in the standard and cluster.

Purposeful Instructional Design: Part IV -- Extend

“Encouraging these practices in students of all ages should be as much a goal of the mathematics curriculum as the learning of specific content” (CCSS, 2010).

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.

Purposeful Instructional Design: The Mathematical Practices

• Teach to the big ideas (clusters and domains).

• Help students bridge concrete, pictorial and abstract representations.

• Provide learning opportunities that require reasoning and sense-making.

• Plan lessons to include 4 important phases: Engage, Explore, Summarize, Extend.

Instruction with Intention

Instruction that Makes a Difference

Effective teachers • reflect on their teaching (focusing on the impact on

student learning), and revise their lessons to more effectively progress students toward achievement of the intended learning target.

• reflect by asking themselves, • Was there opportunity for the students to learn what I

actually intended for them to learn? • What evidence was there that the mathematics was in

fact learned? • What opportunities were missed?