Week 8 Lecture 8

Agenda

• Number Relationship: Part-Part Whole

• Using Fingers in Math

• Composing Trajectory

• Rekenrek

• Homework Check-in

• Phases of Student Learning

• What does fluency with basic facts mean?

• Child Interview

• Homework

Learning Goals

• Understand Part-Part-Whole: Recognize the significance of understanding parts of a number as foundational to number sense. This foundational knowledge helps students grasp how numbers relate to one another and provides the basis for more complex mathematical concepts.

• Using Composing Learning Trajectory: Understand how intentional use can identify developmental benchmarks to enhance instructional decisions, allowing teachers to better tailor their approach to meet individual student needs and provide appropriate challenges that foster growth.

• Basic Fact Reasoning and Fluency: Articulate how fluency with addition and subtraction is rooted in a well-developed number sense. Developing fluency helps students perform operations with confidence and efficiency, which is essential as they progress to more advanced arithmetic and problem-solving tasks.

• Fluency Definition: Comprehend what it means to be fluent with addition and subtraction basic facts, including the ability to quickly and accurately recall the results of these operations, thus enhancing overall mathematics proficiency and reasoning.

Part-Part-Whole Definition

• Focuses on a number's composition in terms of its parts, fostering a deeper understanding of numerical relationships.

• Involves Composition and Decomposition of numbers (e.g., understanding how five can be made from two and three, or how three can be broken down into one and two), which encourages flexible thinking about numbers.

Understanding Standards

• K.OA.A.3: Compose and decompose numbers within 10: Decompose numbers into pairs in various ways to enhance flexibility in number handling.

  • Quickly identify quantities in structured arrangements without counting (conceptual subitizing), a vital skill that supports quicker calculations and numerical understanding.

• K.OA.A.4: Find the complement to make 10 (e.g., determining how much more is needed to reach 10), an important skill for mental math and developing number sense.

Activities to Develop Part-Part-Whole Understanding

Part-Whole Bingo

• Requires gameplay with partners, promoting social interaction and collaborative learning.

• Players roll number cubes to determine how many cubes to cover on their boards, reinforcing the concept of parts within a whole.

• The objective is to cover the entire track (representing the number) completely with cubes. This helps students visualize parts of numbers in a fun context.

• Players need to record what was rolled and how it covered their game board, encouraging metacognitive skills as they reflect on their strategies.

Finger Counting Activities

• Emphasizes the use of fingers for counting and developing numeral sense, tapping into kinesthetic learning styles.

• Encourages students to explore multiple ways to count different sets of numbers using their fingers, fostering creativity in problem-solving.

• Activities can involve patterns of number variation (e.g., counting five in different ways using fingers), helping students visualize and connect different representations of numbers.

Other Activities:

1. Part Whole Mat - Use numeral cards and counters to demonstrate and record different decompositions of numbers, allowing for hands-on manipulation and visual learning.

2. Domino Match - Match numeral cards with dominoes and record the combinations, enhancing visual recognition of quantity relationships.

3. Dot Card Addition - Use dot cards to show pairs that equal a number, including visual representation and written numerals, allowing for a multi-faceted approach to addition.

Gradual Scaffold to Abstract Representation of Quantity

• Progresses from physical representations (like chips and number bonds) to visual representations (like drawings) and finally to abstract reasoning, promoting cognitive development.

• Levels of Understanding: - Physical (using objects),

  • Visual (drawing),

  • Abstract (conceptualizing quantity), which signifies growth in mathematical comprehension.

Missing-Parts Activities

• Activities that focus on determining how many parts are missing from a given number (between 5-10 counters), engaging students in critical thinking.

• Examples include:- Missing Part Baggies: Partners choose a target number, place counters into a baggie, and hide part of them to find the missing part, which fosters problem-solving skills.

  • Snap! Activity: Using a unifix cube train to help students visualize and find missing parts, enhancing their hands-on experience with mathematical concepts.

Composing Trajectory - Learning Levels

• Levels 1-3:

  • Foundations of Parts, building a base for understanding numbers as composites.

  • Part Combiner, learning to combine different parts to form a whole.

  • Inexact Part-Whole Recognizer, developing an initial recognition of number relationships.

• Level 6:

  • Composing quantities using tens and ones, becoming more flexible and strategic in reasoning and problem-solving with quantities, preparing students for complex mathematical challenges in later grades.

Phases of Learning in Basic Arithmetic

1. Direct Modeling: Use objects to model addition and subtraction effectively, allowing for tangible understanding.

2. Counting On: Use visualizations enhanced by counting strategies, which promotes mental math skills.

3. Numerical Reasoning: Apply decomposing and composing strategies to simplify arithmetic operations, equipping students with effective problem-solving tools.

Fluency Development Fundamentals

1. Mastery of Operations: Focus on fluency through varied practice, ensuring students have a robust understanding of mathematical concepts.

2. Practice Phases: Understanding builds through distinct learning stages, emphasizing the importance of progression in mathematical education.

3. Fact Knowledge Base: Prior knowledge must aid the understanding of derived facts, linking past and current learning.

4. Assessment of Fluency: Timing tests may not accurately measure fluency, suggesting the need for more holistic assessment methods.

5. Engaging Practice: Provide enjoyable practice to develop fluency in math, ensuring students remain motivated and invested in their learning journey.

Child Interview Protocol

• Focus: Assessing understanding of counting principles for children aged 4-5, aiding in early detection of learning needs and strengths.

• Deadline for Submission: April 14, 2025 (protocol submission) and April 28, 2025 (final paper submission), ensuring timely completion of assessments.