By Dr. Amy
In his insightful article “The Idea of a Schema” (1987), Richard Skemp introduces us to the concept of ‘intelligent learning.' This approach emphasizes the creation of ‘schemas' – interconnected networks of knowledge that empower students to achieve their learning goals. These schemas, Skemp points out, are not just collections of information; they are dynamic structures that facilitate deeper, more enduring, and versatile understanding.
One of the key insights Skemp shares is the foundational nature of learning: our new knowledge often builds upon what we've already learned. This is particularly true in mathematics. Skemp observed that the more schemas we develop, the better equipped we are to handle new and unexpected challenges.
Central to Skemp's philosophy is the belief that children are capable of intelligent learning from a very young age. As educators, it's our role to nurture and develop their schemas, setting the stage for lifelong learning. A cornerstone of Skemp's theory is the distinction between two types of understanding: relational and instrumental. Relational understanding is about grasping both the ‘how' and the ‘why' of a concept, whereas instrumental understanding is akin to memorizing rules without comprehending their underlying principles.
In simple terms, relational understanding is about truly mastering mathematics, while instrumental understanding is more about rote learning. In his work, “Relational Understanding and Instrumental Understanding,” Skemp delves into the merits of each approach. He notes that instrumental learning, though often quicker and more straightforward, offers short-term gains. On the other hand, relational learning, which may take more time and effort initially, pays off in the long run by fostering adaptability, ease of recall, and self-sustaining growth.
Skemp argues that instrumental understanding has become prevalent, primarily due to its suitability for testing and immediate results. However, he underscores the superior long-term benefits of learning with future applications in mind. While rote learning may yield quick results, it lacks the longevity and depth of relational understanding. Building schemas in mathematics leads to more capable, dynamic learners who find the subject not just manageable but genuinely rewarding.
Skemp also highlights a significant issue in math education: the mismatch between the types of understanding promoted by teachers and desired by students. This discord can lead to frustration and hinder effective learning. For instance, if a student prefers an instrumental approach while the teacher focuses on relational understanding, or vice versa, this can create a challenging learning environment. Skemp points out the less obvious but equally problematic mismatch between teaching methods and educational materials, emphasizing the need for alignment in teaching approaches and resources.
This is where Think! Mathematics textbooks come into play. Rooted in Skemp's theories and those of other educational luminaries like Zoltan Dienes, Jean Piaget, Lev Vygotsky, and Jerome Bruner, these textbooks align with the US Common Core. They are designed to foster relational understanding from an early age. By adopting these resources and teaching mathematics relationally, we equip our students with the foundational knowledge and skills they need to become confident and proficient problem solvers. Embracing these approaches in our teaching can transform the way we educate, leading to more engaged, capable, and enthusiastic mathematicians in our classrooms.