Concreteness Fading: Bridging Procedural and Conceptual Understanding
Mar 1, 2025
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John Jackson, a mathematics education expert, and Emma Lindley-Thompson, a primary education specialist, delve into the intricacies of bridging procedural and conceptual understanding in math. They discuss the challenges of blending concrete and abstract teaching methods, emphasizing the CPA framework and the significance of 'concreteness fading.' The duo also evaluates the effectiveness of manipulatives versus abstract symbols and stresses the importance of teacher professional development to enhance classroom practices, ultimately aiming to boost students' mathematical comprehension.
Integrating procedural knowledge with conceptual understanding in math teaching enhances students' overall comprehension and flexible application of concepts.
Concreteness fading, which transitions from concrete to abstract representations, supports students in better understanding and transferring mathematical knowledge.
Effective teacher professional development is vital for implementing concreteness fading, equipping educators with strategies to connect manipulatives with abstract concepts.
Deep dives
Understanding Procedural and Conceptual Knowledge
Procedural knowledge refers to the ability to perform specific mathematical operations, while conceptual understanding involves grasping the underlying principles behind those operations. For example, when calculating the area of a triangle, knowing the formula involves procedural knowledge, but understanding why the formula works requires conceptual insight. This spectrum indicates that students can possess varying depths of understanding, and the successful integration of both types of knowledge is crucial for flexible application of math concepts. It's essential to recognize that emphasizing one over the other in teaching can have significant implications on students' comprehension and performance.
The Debate on Teaching Approaches
There has been ongoing debate regarding the balance between teaching procedural knowledge and conceptual understanding in mathematics. Some educators argue that students should focus on practicing procedural tasks until they master them, while others advocate for an understanding of concepts before procedures are taught. Research has shown that integrating both elements can lead to improved learning outcomes since they reinforce each other. This iterative approach encourages educators to toggle between teaching concepts and procedures, thereby enhancing students' overall mathematical understanding.
The Importance of Concreteness Fading
Concreteness fading is a teaching technique where instruction starts with concrete representations before gradually transitioning to abstract symbols. This method has been found to support students in developing a solid understanding of mathematical concepts while still allowing them to utilize efficient abstract reasoning. The research indicates that when students engage with concrete materials, they are better equipped to transfer their knowledge to new, abstract problems. This pedagogical approach advocates for a structured progression from concrete to abstract representations, helping students internalize and generalize their mathematical skills.
Choosing Effective Representations
Selecting the right manipulatives and representations is critical to effective math instruction, as it influences how students understand and apply mathematical concepts. Teachers are encouraged to use a combination of physical objects, pictorial representations, and abstract symbols tailored to the content being taught. For instance, bar models can bridge the gap between concrete and abstract understandings, but their effectiveness depends on making explicit the connections between the different forms of representation. This choice should also consider the students’ prior knowledge and the specific mathematical concepts at hand, ensuring that the learning process is meaningful and coherent.
Implementing Concreteness Fading in Professional Development
Teacher professional development should focus on the effective implementation of concreteness fading strategies, providing educators with opportunities to practice using manipulatives and gradual transitions in classroom settings. This involves equipping teachers with the necessary knowledge about how to utilize concrete representations effectively while connecting them to abstract concepts. Educators should have the chance to explore various manipulatives and reflect on their appropriateness for different mathematical contexts. Ongoing support and collaboration among teachers can enhance the successful integration of these strategies, leading to improved student learning outcomes in mathematics.
This week on Thinking Deeply about Primary Education, John Jackson and Emma Lindley-Thompson from Ambition Institute return to explore the next extract from their expert edit—this time, focusing on concreteness fading and its role in mathematics teaching.
We discuss:
The relationship between procedural knowledge and conceptual understanding and how this influences mathematics instruction.
The challenge of premature reliance on abstract symbols and how teachers can strike the right balance between concrete and abstract representations.
The debate over concrete representations—do they make abstract ideas clearer, or do they hinder transfer to new contexts?
Key implications for teacher professional development and how educators can implement concreteness fading effectively in the classroom.
Whether you’re a maths specialist, a primary educator, or a leader supporting professional development, this episode provides valuable research-backed insights to enhance mathematical understanding in your teaching.
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