In schools, different teachers prefer different teaching approaches. Some teachers emphasise learning for understanding while others, by contrast, tend to emphasise memorisation.
Within Kilpatrick, Swafford and Findell’s (2001) Strands of Mathematical Proficiency, learning with understanding is referred to as conceptual understanding while memorisation of mathematical facts and procedures fall under the banner of procedural fluency.
Conceptual understanding focuses on the understanding of mathematical concepts and relations, thereby helping learners to make connections between concepts and to relate them to one another.
This also resonates with Hiebert and Lefevre (1986, pp. 3-4) who describe conceptual knowledge as “a connected web of knowledge, a network in which the linking relationships are as prominent as the discrete pieces of information.”
In Namibia, the Ministry of Education has embraced Learner-Centred Education (LCE) as an approach for teaching and learning. LCE resonates strongly with the notion of conceptual understanding as it emphasises learning with understanding. In learner-centred education the role of the teacher is to assist, motivate and facilitate learning, and learners are expected to take ownership of their learning.
Conceptual understanding and Learner-Centred Education resonate strongly with each other and all emphasize that classrooms that foster a spirit of developing learners’ conceptual understanding are characterised by, amongst other things, a focus on why and how as opposed to what, multiple solution strategies, transferring concepts to novel contexts, an emphasis on connections, the use of multiple representations or models, building on prior knowledge, connecting mathematics to the real world, and questioning that promotes critical thinking.
Conceptual understanding also emphasises that learning should be connected to what learners already know. Teaching which ignores and does not build on learners’ prior experience and learning will limit learners’ thinking as they are less likely to see the connection between the world outside school and what is taught and learnt in the school.
Additionally, conceptual understanding suggests that the construction of knowledge is shaped by the social environment and teachers need to structure their teaching on the interactions between learners. It is, therefore, important that teachers, as the facilitators of knowledge in the classroom, should help learners make connections between existing knowledge and learned knowledge. They can do so by guiding learners from what they currently know to what they are about to learn.
Moreover, teachers can assess conceptual understanding by asking questions such as “why?” and “how?”, asking learners to solve and represent mathematical problems in different ways, or asking questions that require critical thinking. In addition, classrooms that incorporate problem solving promote conceptual understanding in learners because learners are involved in high levels of reasoning and critical thinking. The teacher should also prepare activities that prompt learners’ experience and help them to construct knowledge.
Andreas A. Kashima
Omusati Region