Chapter 9

Value Concepts and Strategies

for

Fact Fluency

VandeWalle, et al., 2019. p.184

Learning Outcomes Aligned for Teaching Basic Facts

1. Describe the Developmental Process and Research-Based Approach for Teaching Basic Facts

Focus on the sequential development from counting, to using strategies, to achieving automaticity.

Counting: Begin with teaching students to count objects and numbers, laying the foundation for understanding basic facts.

Using Strategies: Introduce various strategies to make connections and solve problems, such as decomposing numbers, using doubles (e.g., 7 + 7), and making tens (e.g., 8 + 6 = 8 + 2 + 4).

Achieving Automaticity: Guide students toward automaticity where they can recall basic facts effortlessly and quickly without needing to count or use strategies.

Emphasize understanding over mere memorization.

Conceptual Understanding: Ensure students understand the "why" behind the facts. This deeper comprehension leads to better retention and application of knowledge.

2. Strategies for Developing Addition and Subtraction Fact Fluency

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Decomposing Numbers: Teach students to break down numbers into smaller, more manageable parts (e.g., 8 + 6 can be thought of as 8 + 2 + 4).

Using Doubles: Leverage facts that students already know (e.g., 7 + 7) to solve related problems.

Making Tens: Encourage students to make combinations of ten, which simplifies addition and subtraction (e.g., 9 + 1, 8 + 2).

3. Strategies for Developing Multiplication and Division Fact Fluency

Using Known Facts: Help students use facts they know to solve related problems (e.g., knowing 6 × 7 helps with 6 × 8 by understanding the relationship).

Grouping: Teach students to understand multiplication and division through grouping (e.g., 3 groups of 4).

Arrays or Area Models: Use visual models to illustrate multiplication and division, making abstract concepts more concrete.

4. Justify Limitations of Timed Tests and Explain Effective Assessment Alternatives

Critique of Timed Tests: Highlight the stress and anxiety timed tests can cause, which may lead to rote memorization without understanding.

Effective Alternatives: Advocate for assessments that focus on reasoning and understanding, such as:

Problem-Solving Tasks: Assess students' ability to apply their knowledge in different contexts.

Performance Assessments: Evaluate students through activities that require explanation and demonstration of their understanding.

5. Effective Methods for Ongoing Practice and Intervention for Basic Fact Fluency

Ongoing Practice:

Games and Activities: Incorporate engaging, hands-on activities that reinforce basic facts.

Real-World Problem-Solving: Use real-life scenarios to practice facts in a meaningful context.

Intervention Strategies: Identify specific areas where students struggle and tailor interventions to address these gaps, providing additional support as needed.

Definition of Basic Facts: Basic addition and multiplication facts where both addends/factors are less than 10; subtraction and division facts are their corresponding combinations.

Teaching Approach: Focus on developing fluency (speed with accuracy) and automaticity (effortless recall) through reasoning strategies rather than rote learning.

Development Phases: Support students as they progress from counting to using strategies and ultimately achieving automaticity, ensuring instruction is paced appropriately.

Strategies: Promote understanding by leveraging known relationships (e.g., doubles, making tens) rather than relying solely on memorization.

Benefits: Emphasize building strong number sense and confidence, which supports broader mathematical proficiency beyond mere memorization.

Three Phases of Fluency Development:

Counting: The initial phase where students rely on counting objects or numbers to find answers.

Using Strategies: Students learn and apply various strategies to solve problems more efficiently.

Achieving Automaticity: The final phase where students can recall basic facts quickly and effortlessly.

Instruction and assessment must guide students through these phases at a comfortable pace, ensuring they fully understand each step before moving on.

Strategic Understanding of Number Relationships:

Teaching students to understand and use number relationships provides a strong foundation for fluency.

Example: For solving 8 + 6, students can decompose 6 into 2 and 4, then add the 2 to the 8 to make a 10, and finally add the remaining 4 to get 14. This is known as the "Make 10" strategy.

By focusing on these big ideas, educators can support students in developing a deeper understanding and fluency with basic facts, laying the groundwork for more advanced mathematical concepts.

1. Developmental Process and Research-Based Approach:

Students move through three phases in developing fluency with basic facts: counting, using strategies, and achieving automaticity. Instruction and assessment must help students progress through these phases without rushing them.

2. Understanding Over Memorization:

The goal for basic facts is to achieve efficiency and accuracy through understanding and reasoning, not just memorization.

3. Strategies for Developing Fact Fluency:

Addition and Subtraction: Use strategies such as decomposing numbers, using doubles (e.g., 7 + 7), and making tens (e.g., 8 + 6 = 8 + 2 + 4).
Multiplication and Division: Employ strategies like using known facts, grouping, and using arrays or area models.

4. Limitations of Timed Tests:

Timed tests can cause stress and promote rote memorization without understanding. Effective alternatives include problem-solving tasks and performance assessments that focus on reasoning and understanding.

5. Ongoing Practice and Intervention:

Encourage ongoing practice through games, activities, and real-world problem-solving. Use intervention strategies to identify and address specific areas where students struggle.

Phase 1: Counting Strategies:

Students use objects or verbal counting to determine answers.

Example: For 4 + 7, the student counts on from 7 (8, 9, 10, 11). For 4 × 7, the student skip counts by 7s (7, 14, 21, 28).

Phase 2: Reasoning Strategies:

Students use known information to logically determine unknown combinations.

Example: For 4 + 7, the student knows that 3 + 7 equals 10, so 4 + 7 is one more, 11. For 4 × 7, the student knows 2 × 7 equals 14 and doubles that to get 28.

Phase 3: Mastery or Automaticity:

Students can quickly and effortlessly recall facts.

Example: For 4 + 7, the student quickly responds, "It's 11." For 4 × 7, the student automatically knows it's 28.

When Are Students Ready to Work on Reasoning Strategies?

Readiness Indicators:

Students can use counting-on strategies (start with the largest number and count up).

Students can decompose numbers within 10 (e.g., decompose 6 into 5 + 1).

Quick Interview Questions to Determine Readiness:

Counting On:

Ask: 7 + 2, 1 + 6, 10 + 2, 2 + 9

Check: Do students count on from the larger number or just know the answer?

Decomposing:

Ask: How can 6 children sit at two tables?

Probe: Look for more than one way to split the children.

Addition Expressions:

Ask: Write addition expressions that equal 7.

Observe: Can students come up with different combinations?

Assessment Approach:

Ask these questions informally during activities.

Set up a station for students to rotate through and answer these questions.

Approaches to Teaching Basic Facts:

Memorization Approach:

Skips the development of reasoning strategies.

Inefficient due to the large number of facts to memorize (100 addition, 100 multiplication, and their corresponding subtraction and division facts).

Results in inflexibility and inappropriate application of facts.

Leads to lower number sense and a dislike for math.

Explicit Strategy Instruction:

Focuses on teaching and practicing strategies (e.g., Make 10, doubles).

Supported by extensive research as more effective than memorization.

Helps students understand and remember basic facts.

Builds thinking and problem-solving skills, encouraging students to use various strategies.

Why Explicit Strategy Instruction Works:

Efficiency: Reduces the number of facts to memorize by using strategies.

Flexibility: Equips students with multiple strategies to solve problems.

Application: Helps students check their work and understand why their answers are correct.

By focusing on explicit strategy instruction, educators can improve students' number sense, problem-solving skills, and overall confidence in math.

Developmental Approach to Teaching Basic Facts:

Teaching basic facts involves helping students progress through counting, reasoning strategies, and achieving automaticity.

Use subsets of facts in an appropriate sequence to support this progression.

Assessing Student Strategies:

When working on facts, ask students how they solved problems like 7 + 4.

Identify whether they are counting (Phase 1) or using reasoning strategies like "Make 10" (Phase 2).

Encourage those who count to count on, and those using strategies to continue applying them.

Engaging Activities:

Use number talks, basic fact games, and diverse experiences to help students develop and apply strategies.

Over time, students will become quicker and more automatic in recalling facts.

Story Problems:

Emphasize story problems to improve problem-solving skills and automaticity.

Pose a story problem daily and discuss the strategies used.

Example: "Rachel had 9 ponies in one barn and 6 ponies in the pasture. How many ponies did she have altogether?" This helps students think of 9 + 6 as 10 + 5.

Paired Story Problems:

Prepare two corresponding stories: one with a foundational fact and another requiring a reasoning strategy.

Example:

Story 1: "In 2 weeks, we will go to the zoo. How many days until we go to the zoo?"

Story 2: "In 4 weeks, we will go to a farm. How many days until we go to the farm? Use your thinking from Story 1 to help solve this problem."

Have students solve the first problem, then use their solution to help solve the second.

Key Benefits of Using Strategies:

Helps students understand the context and apply flexible strategies.

Supports deeper understanding and accurate computation.

Promotes the use of known facts to solve new problems.

Purpose of Quick Looks:

Move students beyond counting to understanding how numbers can be composed or decomposed.
Highly effective routine for young children (McCoy, Bame, & Combs, 2013; Shumway, 2011).

Tools for Quick Looks:

Use dot cards, five-frames, and ten-frames to help students see number relationships.
Ten-frames are particularly useful for developing strategies and understanding number relations.

Quick Look Routine:

Step 1: Flash an image (e.g., Quick Image 1) for a few seconds, then hide it.
Step 2: Repeat Step 1 with the same image.
Step 3: Show the image a third time and ask, "How many did you see?" Collect answers.
Step 4: Ask, "How did you see it?" Listen to different ways students saw the total. Draw or point to the aspects described for all students to see.

Pairing Images:

Pair two images to help students see relationships that lead to strategies.
Example: Use paired images to highlight the "Make 10" strategy or doubling.
Repeat the full quick look routine with each image, then compare them side by side in Step 4 to see relationships.

Patterns and Strategies:

Conduct quick looks for three to four pairs of images during a number talk.
Conclude by asking students what patterns they noticed and how these patterns can help solve basic fact problems.

Example Activities:

Addition: Pair images to show combinations of 10 (e.g., 7 + 3, 8 + 2).

By using the quick look routine, you can help students develop a deeper understanding of number relationships and strategies for solving basic facts.

Multiplication: Pair images to show doubling strategies (e.g., 2 x 3, 4 x 3)

Addition and Subtraction Facts

Strategies for Addition and Subtraction

Use physical objects like counters, blocks, or beads to represent addition problems.
Use to drawings or pictures that represent the addition problems.
Use numbers and symbols.

Number line

Use number lines to visually show how numbers are added. Students can physically move along the line to see how numbers combine.
Use number lines to visually show how numbers are subtracted. Students can physically move backward along the line to see how numbers decrease.

Counting On/ Counting Backwards

Teach students to start with the larger number and count on the smaller number. For example, for 5 + 3, start at 5 and count 6, 7, 8.
Teach students to count backward from the larger number. For example, for 7 - 3, start at 7 and count 6, 5, 4.

Ten Frames

Ten frames help students see how numbers combine to make ten, which is a key concept in understanding addition.
Ten frames help students see how numbers are taken away from ten, which is essential for understanding subtraction

Fact Families

Teach addition and subtraction together using fact families. For example, if students know 3 + 4 = 7, they also know 4 + 3 = 7, 7 - 4 = 3, and 7 - 3 = 4.

Games and Activities

Flashcards: Use flashcards for quick recall of addition and subtraction facts.
Addition Bingo: Create bingo cards with sums and call out addition problems.
Subtraction Bingo: Create bingo cards with differences and call out subtraction problems.
Online Games: Utilize educational websites and apps that offer interactive addition games.

Word Problems

Use real-life scenarios to create word problems. This helps students understand the practical application of addition.
Encourage and praise students for their efforts and progress. It boosts their confidence and motivation.

Manipulatives and Games

Teaching Multiplication and Division

1.Use of Manipulatives and Visual Aids

Arrays and Area Models: Use objects like counters, blocks, or grid paper to create visual arrays.
Number Lines: Use number lines to help students visualize multiplication and division. For multiplication, they can make equal jumps, and for division, they can see how many times one number fits into another.
Visual Aids: Charts and diagrams, such as multiplication tables and fact triangles, can help students see relationships between numbers.

2. Games and Interactive Activities

Flashcards: Use multiplication and division flashcards for quick recall practice.
Online Games: Interactive math games can make learning fun and engaging.
Board Games: Games like “Multiplication Bingo” or “Division War” can help reinforce facts in a fun setting.

3. Story Problems and Real-Life Contexts

Word Problems: Incorporate multiplication and division into story problems to help students understand their practical applications.
Real-Life Scenarios: Use scenarios such as dividing snacks among friends or multiplying ingredients in a recipe to provide context.

Here is a virtual link to a free online base 10 manipulation:

https://oryxlearning.com/manipulatives/base-ten-blocks

Accessing Basic Fact Fluency

Addition/Subtraction

What's so important about fact fluency?

Fact fluency is crucial for mathematical proficiency and involves quick and accurate recall of basic addition, subtraction, multiplication, and division facts. Basic fact fluency involves the quick and accurate recall of basic math facts, such as addition, subtraction, multiplication, and division. It's important because it helps students tackle more complex problems with ease. Let's dive into different methods for assessing basic fact fluency. First, we have oral assessments. These are quick interviews where you ask students to solve problems verbally. Next, written assessments, such as timed tests and worksheets, can give you a clear picture of a student's speed and accuracy. Finally, digital tools like educational apps can make assessments fun and engaging for students. Look for patterns in accuracy and speed. Set benchmarks to track progress over time and identify areas where students may need additional practice. To make assessments more effective, keep them low-stress and integrate them into your daily routine. For example, start each math lesson with a quick five-minute fluency drill.

How can you help your students master fact fluency?

- Counting Strategies: Using fingers or objects to count.
- Reasoning Strategies: Using known facts to derive unknown ones (e.g., using doubles or making ten).
- Mastery: Instant recall of facts without effort.

Useful strategies:

- Use of Manipulatives: Tools like counters, number lines, and arrays to visualize math facts.
- Games and Activities: Engaging students with math games that reinforce fact recall in a fun and interactive way.
- Drill and Practice: Short, frequent practice sessions that focus on a small set of facts until mastery is achieved.
- Fact Families: Teaching related facts together to help students see the relationships between them (e.g., 3 + 4 = 7 and 7 - 4 = 3).

How should you teach basic fact fluency?

- Guided Discovery: Encouraging students to find patterns and relationships in math facts through exploration and discussion.
- Peer Tutoring: Pairing students to practice facts together, fostering both learning and teaching skills.
- Technology Integration: Using educational software and apps that provide immediate feedback and adapt to the student’s level.

Example and Instruction:

Example: To teach addition facts up to 10, start with counting objects. Progress to using a number line, then move to games like "Addition War" where students flip cards and add the numbers. Finally, use timed drills for rapid recall.
Instruction: Introduce the concept with manipulatives, model the use of reasoning strategies, provide ample practice with varied activities, and assess regularly to ensure mastery before moving on.

AdditionSubtractionMathFactsFluencyComputationDailyPractice2ndGrade-1.pdf

Daily Fluency Practice

5inaLineAdditionSubtractionFactFluency-1.pdf

5 in a Line Game

AdditionandSubtractionFactFluencyLogs-1.pdf

Fluency Logs

MathFluencyFacts-1.pdf

Fluency Flashcards

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