The concrete, pictorial, and abstract (CPA) teaching approach is a widely used method to help students understand and internalize mathematical concepts. Here’s how you can authentically integrate this approach:

1. Start with concrete materials: Begin by introducing a concrete representation of the concept you are teaching. This could be manipulatives, objects, or real-life examples that students can physically interact with. For example, if you are teaching addition, you can use counters or blocks for students to physically combine and count.

2. Move to pictorial representations: Once students have a solid understanding of the concept through concrete materials, transition to pictorial representations. This involves using drawings, diagrams, or visual representations to represent the concept. For example, you can use pictures or drawings to represent the addition of numbers instead of physical objects.

3. Progress to abstract representations: After students have grasped the concept through concrete and pictorial representations, move towards abstract representations. This involves using numbers, symbols, and equations to represent the concept. For example, you can introduce addition using numbers and mathematical symbols without the need for physical objects or pictures.

4. Provide opportunities for practice and application: Throughout the process, provide ample opportunities for students to practice and apply their understanding of the concept. This can include solving problems, engaging in hands-on activities, and participating in group discussions. Encourage students to explain their thinking and reasoning using all three representations (concrete, pictorial, and abstract).

5. Scaffold and differentiate instruction: Different students may require different levels of support and scaffolding. Some students may need more time with concrete materials, while others may quickly progress to abstract representations. Differentiate your instruction by providing additional support or challenges based on individual student needs.

6. Make connections to real-life situations: Whenever possible, relate the mathematical concepts to real-life situations. This helps students see the relevance and applicability of what they are learning. For example, when teaching fractions, you can use real-life examples like dividing a pizza or sharing a cake.

7. Continuously assess and provide feedback: Regularly assess students’ understanding of the concept and provide timely feedback. This can be done through formative assessments, observations, and discussions. Use the feedback to guide your instruction and address any misconceptions or gaps in understanding.

Remember, the key to authentically integrating the CPA approach is to provide a gradual progression from concrete to pictorial to abstract representations, while also considering the individual needs and learning styles of your students.