Lecture 9 Powerpoint

Agenda

• Phases of Student Learning

• State Standards for fluency with addition and subtraction facts

• Meaning of fluency with basic facts

• Strategies for addition basic facts

• Homework

Learning Goals

• Articulate the relationship between basic fact reasoning, fluency, and number sense by exploring how understanding number relationships enhances comprehension and application of mathematical concepts.

• Understand what fluency entails for addition and subtraction basic facts, including the ability to recall facts quickly and accurately, as well as the understanding of underlying mathematical principles.

• Develop awareness of the WI State Standard expectations for fluency, which emphasize not just the speed of recall, but the conceptual understanding that supports fluent problem-solving.

• Learn various strategies for addition basic facts that encourage deep thinking, reasoning, and application of learned skills in different contexts.

Key Concepts in Understanding Addition and Subtraction

Learning Parts of Numbers (pp. 48-51)

1. Containment: Recognizing that one number can be part of another (e.g., 5 is composed of 3 and 2) aids in understanding how numbers interact within operations.

2. Decomposition: The ability to break down numbers in various ways (e.g., 5 can be expressed as 3+2, 4+1, or 5+0), which enhances flexibility in problem-solving.

3. Commutative Property: Chunks of numbers can be combined in any order to simplify addition.

   • Example: (8 + 6 = 14) can be simplified by first combining (8 + 2) to make 10 and then adding 4.

Identifying Parts of Numbers (pp. 52-55)

1. Children must move from viewing numbers as collections of individual units to understanding the relationships between these units.

2. Different representations of quantities (like using manipulatives, drawings, and symbols) help children recognize parts contained within larger numbers, facilitating deeper comprehension.

3. Subitizing: Recognizing quantities through visual patterns supports numerical understanding without needing to count.

Basic Facts and Understanding Math Relationships (pp. 43-55)

1. Knowledge of basic math facts should be rooted in an understanding of how numbers compose and decompose, connecting to larger mathematical ideas.

2. Merely memorizing facts without understanding their relationships does not constitute real understanding (similar to knowing a name but not knowing the person).

Strategies for Addition

Example: Adding 8 + 6

• Direct Modeling: Using counters or drawings to represent addition step by step helps visual learners understand the process.

  • Example: Count up from 8 or visualize the addition through physical objects or drawings.

• Counting On: Start with the first number and count up the second, which aids in building number sense and reinforces addition concepts.

  • Example: "I start at 8 and count up 6 (9, 10, 11, 12, 13, 14)".

• Make 10 Strategy: Breaking down numbers to make 10 simplifies addition.

  • Example: 8 + 6 = 8 + (2 + 4) = (8 + 2) + 4 = 10 + 4 = 14.

• Using Doubles: Recognizing that if 6 + 6 = 12, then to add 8, we can calculate 12 + 2 = 14, reinforcing the concept of number relationships.

• Using Fives: Segmenting numbers into groups of 5 enhances mental calculation strategies.

  • Example: 8 + 6 = (5 + 3) + (5 + 1) = 10 + 4 = 14.

Phases of Learning Addition and Subtraction

1. Phase 1: Direct Modeling - Children use objects, pictures, and counting to model addition and subtraction, fostering a tactile connection to mathematical concepts.

2. Phase 2: Counting On - Visualizing quantities and counting on from one of the addends supports numerical understanding and fluency.

3. Phase 3: Numerical Reasoning - Using number relationships and strategic reasoning to solve problems allows learners to apply concepts in varied contexts.

Fundamental Concepts of Fluency

Five Fundamentals of Fluency

1. Fluency Development: Fluency develops through clearly defined phases that encourage depth of understanding and mastery.

2. Mastery Focus: Emphasis must be placed on true fluency rather than rote memorization, ensuring that students grasp the why behind calculations.

3. Foundational Facts: Understanding basic facts paves the way for students to grasp derived facts, fostering advanced problem-solving skills.

4. Testing: Timed tests do not accurately assess fluency; evaluations should focus on understanding and application.

5. Practice: Students require ample enjoyable practice to develop fluency, integrating real-world applications to illustrate mathematical concepts.

Expected Fluency by State Standards

• Kindergarten: Master addition and subtraction within 5, establishing foundational skills for future learning.

• Grade 1: Work with numbers up to 20, increasing complexity and building confidence in numerical operations.

• Grade 2: Extend operations and problem-solving to 100, ensuring a solid grasp of addition and subtraction as foundational skills for higher-level mathematics.

Homework and Collaboration

• Discuss key points and insights from readings at your tables to reinforce learning. Be prepared to share insights on addition and subtraction strategies to enhance collaborative learning experiences.

Games for Building Fluency

• Utilize interactive games like "Roll and Total" and "Make 10" to promote reasoning and fluency in engaging ways.

• Each game targets specific skills, supporting fluency development through enjoyable practice and contextual application.