5.3: John Van de Walle's Strategy

John Van de Walle was an influential mathematics educator whose work significantly impacted how mathematics is taught in schools. Born in 1944, Van de Walle was a professor emeritus at Virginia Commonwealth University, where he dedicated his career to improving mathematics education. He is best known for his book "Elementary and Middle School Mathematics: Teaching Developmentally," which offers a comprehensive guide for teachers on how to develop students' mathematical thinking. Van de Walle emphasized the importance of understanding mathematical concepts rather than simply memorizing procedures. His approach to problem-solving focuses on developing students' ability to think critically and solve problems in a way that is meaningful and connected to their experiences.

Van de Walle’s problem-solving strategy is centered around fostering a deep understanding of mathematical concepts through active engagement and exploration. His approach involves encouraging students to explore multiple strategies, make connections, and reflect on their thinking. By promoting a classroom environment where students are encouraged to discuss and share their ideas, Van de Walle's strategy helps students develop a robust understanding of mathematics. His methods are designed to be student-centered, focusing on the development of problem-solving skills through real-world applications and collaborative learning.

Van de Walle's Problem-Solving Strategy

1. Understand the Problem:
   • Engage students in the problem: Use real-world contexts that are meaningful to students to capture their interest.
   • Clarify the problem: Ensure that students understand the problem by discussing it and rephrasing it in their own words.
   • Identify what is known and unknown: Help students distinguish between the given information and what needs to be determined.
2. Explore and Plan:
   • Encourage multiple strategies: Promote the use of various strategies to solve the problem, such as drawing diagrams, using manipulatives, or looking for patterns.
   • Facilitate discussion and collaboration: Create an environment where students feel comfortable sharing their ideas and strategies with peers.
   • Guide students to make connections: Help students relate the current problem to previously learned concepts and experiences.
3. Implement the Plan:
   • Support students as they work: Provide guidance and encouragement as students carry out their chosen strategies.
   • Monitor progress and provide feedback: Observe students’ work and offer constructive feedback to help them stay on track.
   • Encourage perseverance: Foster a growth mindset by encouraging students to keep trying even if they encounter difficulties.
4. Reflect and Discuss:
   • Review the solution: Discuss the solution as a class, highlighting different strategies used and their effectiveness.
   • Reflect on the process: Encourage students to think about what they learned and how they approached the problem.
   • Generalize learning: Help students apply the problem-solving strategies they used to new and different problems.

John Van de Walle’s problem-solving strategy emphasizes understanding, exploration, and reflection, making it a powerful approach for developing students' mathematical thinking. By focusing on real-world contexts, multiple strategies, and collaborative learning, Van de Walle’s methods encourage students to engage deeply with mathematical concepts. His approach not only helps students solve individual problems but also fosters a broader understanding of mathematics that can be applied to various situations. Van de Walle’s contributions to mathematics education continue to influence teaching practices and inspire educators to cultivate a love for mathematics in their students.

Example \(\PageIndex{1}\)

Problem:

Sarah has a rectangular prism-shaped box that she wants to fill with small 1-unit by 1-unit by 1-unit cubes. The dimensions of the box are 5 units long, 3 units wide, and 4 units high. How many 1-unit cubes will Sarah need to completely fill the box?

Solution

Step 1: Understand the Problem

• We need to find the total number of 1-unit by 1-unit by 1-unit cubes required to fill a rectangular prism.
• The dimensions of the rectangular prism are given as 5 units (length), 3 units (width), and 4 units (height).

Here's how a teacher might approach this step:

1. Introduce the Concept: Begin with a discussion about volume and three-dimensional shapes. Use real-life examples, such as filling a box with cubes, to make the concept relatable.
2. Clarify the Task: Explain the problem clearly, ensuring that students understand the dimensions given (length, width, height) and the goal (filling the box with unit cubes).
3. Ask Guiding Questions: Prompt students with questions like, "What are we trying to find?" and "What information do we have?" to ensure they comprehend the problem.
4. Use Visuals: Provide diagrams or physical models of rectangular prisms to help students visualize the problem.

Step 2: Developing and Using Strategies

• To find the volume of the rectangular prism, we can use the formula for the volume of a rectangular prism:
  \(Volume = Length \times Width \times Height\)

• Here, the length is 5 units, the width is 3 units, and the height is 4 units.

Here's how a teacher might approach this step:

• Discuss Possible Strategies: Encourage students to brainstorm different strategies to find the volume. Discuss the formula for the volume of a rectangular prism and how it relates to multiplying the dimensions.
• Model the Process: Demonstrate how to apply the volume formula using a step-by-step approach. Show how each dimension is multiplied.
• Encourage Estimation: Ask students to estimate the volume before calculating it to develop their estimation skills and understanding of the problem's scale.
• Collaborative Learning: Allow students to work in pairs or groups to discuss and develop their strategies, fostering collaboration and peer learning.

Step 3: Implementing the Solution

• Using the volume formula:
  \(Volume = 5 units \times 3 units \times 4 units\)
  \(Volume = 60 \ cubic \ units \)
• Sarah needs 60 small 1-unit by 1-unit by 1-unit cubes to completely fill the box.

Here's how a teacher might approach this step:

• Guide the Calculation: Walk students through the calculation process. Ensure they understand each step and how it contributes to finding the final volume.
• Use Manipulatives: Provide unit cubes for students to physically fill a model of the box, reinforcing the concept of volume and the multiplication of dimensions.
• Check for Understanding: Frequently check in with students to ensure they are following along and comprehending each step. Ask questions to probe their understanding.

Step 4: Reflecting on the Process and Solution

• We checked that we used the correct formula for volume and applied the given dimensions correctly.
• We ensured that the solution (60 cubic units) makes sense for the given problem.

Here's how a teacher might approach this step:

• Encourage Reflection: After solving the problem, ask students to reflect on the process. Questions like, "What steps did we take to solve the problem?" and "Why did we use this method?" help deepen their understanding.
• Discuss Errors: If mistakes were made, discuss them openly. Analyze why the errors occurred and how to correct them, turning mistakes into learning opportunities.
• Connect to Real Life: Discuss how this problem-solving approach can be applied to other real-world scenarios, reinforcing the practical value of the concept.

Step 5: Building Conceptual Understanding

• This problem helps students understand that the volume of a three-dimensional shape can be determined by multiplying its dimensions.
• They also see that the volume represents the number of unit cubes that can fill the shape without gaps or overlaps.
• Understanding this concept is crucial for recognizing how to measure and calculate volume in various real-world and mathematical problems.

Here's how a teacher might approach this step:

• Revisit the Concept: Regularly revisit the concept of volume and its measurement in different contexts to reinforce understanding.
• Use Varied Examples: Provide different problems involving volume, including irregular shapes and real-world applications, to deepen students' conceptual understanding.
• Encourage Exploration: Allow students to explore and experiment with different three-dimensional shapes and their volumes, fostering curiosity and deeper learning.
• Assess Understanding: Use formative assessments, such as quizzes or quick checks, to gauge students' understanding and provide feedback to support their learning journey.

Example of Implementing Each Step:

1. Understanding the Problem:
   • Teacher: "Today, we’re going to solve a problem involving the volume of a box. Can anyone tell me what volume means?"
   • Students: "It's how much space is inside an object."
   • Teacher: "Right! We have a box that is 5 units long, 3 units wide, and 4 units high. We need to find out how many small cubes will fit inside this box."
2. Developing and Using Strategies:
   • Teacher: "What strategies can we use to find the volume of this box?"
   • Students: "We can multiply the length, width, and height."
   • Teacher: "Exactly! The formula for volume is length times width times height. Let's apply this formula."
3. Implementing the Solution:
   • Teacher: "Let’s calculate the volume together. What is 5 times 3?"
   • Students: "15."
   • Teacher: "And 15 times 4?"
   • Students: "60."
   • Teacher: "So, the volume is 60 cubic units. This means we need 60 small cubes to fill the box."
4. Reflecting on the Process and Solution:
   • Teacher: "Why did we multiply these numbers? What do each of these numbers represent in our problem?"
   • Students: "They represent the dimensions of the box."
   • Teacher: "Yes, and multiplying them gives us the total volume. How can we check our work to make sure it’s correct?"
5. Building Conceptual Understanding:
   • Teacher: "Can you think of other objects that we could measure the volume of? How would you approach finding their volume?"
   • Students: "We could measure the volume of a fish tank or a carton of milk by using the same formula."
   • Teacher: "Great! Remember, understanding how to find volume helps us in many real-world situations."

By following these steps and approaches, teachers can effectively guide students through solving volume problems and help them build a solid understanding of three-dimensional measurement.