Concept-Based Learning in the Mathematics Classroom: Tools for the Teacher
In this section, I share and elaborate some of the strategies that encourage deep learning in students and tackle some of the challenges faced by the teacher in a concept-based learning environment as listed above, including the problems related to the transfer of knowledge.
Tool 1: Teaching Mathematics Conceptually
The classroom teacher in the concept-based learning environment is not merely delivering content. The most typical scene in the classroom today is that concepts are taught mostly through deductive methods, where a new theorem or a new formula is introduced before students get to know how they are applied in examples. This takes away the opportunity for students to develop their own understanding. To teach conceptually means that the teacher would be acting as a facilitator to help and guide students to construct knowledge and concepts to encourage deep links between what they know and what they are learning. The learning environment of the class must be one of open-mindedness, where every individual has a chance (prompted if need be) to be heard, so that there are opportunities for anyone to clarify and to learn from one another’s perspective. When the environment is well set up to promote learning and teaching, it is then up to the teacher to pace the lesson such that the concepts are delivered in a manner which is engaging to the whole class.
To initiate concept-based learning in the classroom, it is important to know the readiness of every individual, including his learning style. Although this is easier said than done, given a classroom of 25-35 students of varying abilities and giftedness, the most common mistake, in my experience, is that when teachers assume that every student in the classroom has the same background knowledge, ability and learning style. Hence, the challenge to the classroom teacher would be to ensure that every individual is engaged in his or her own zone of proximal development, where instruction can be differentiated. Hence, equipped with such nuanced knowledge of the learner, the highly gifted and able can be provided with enriched or advanced materials to deepen their understanding of a topic or concept in their own time while the rest of the class is given the time to figure out the foundational concepts at their own pace with the teacher. Most teachers would have realised that gone are the days when students sit passively listening to the teacher standing at the front lecturing continuously, especially given competition from digital devices. Disengaged learners and teacher-driven lecturing in front of the whole class do not promote conceptual understanding of the discipline.
A good start to a concept-based mathematics lesson would be to begin with a recap of the previous concepts that might form the prerequisites for the lesson or to introduce a scenario where students do not yet know how to solve or even understand the problem. Students can then begin to recall previous concepts learnt. Sometimes, it may be necessary for the teacher to spend a considerable amount of time revising the previous concepts if it is found that (a) the students may not have yet fully understood the prerequisites or (b) there are several sub-concepts involved prior to learning new ones.
It is important to provide sufficient scaffoldings in concept-based learning. From Vygotsky’s theory of ZPD, we learn that the teacher must provide sufficiently challenging tasks for the student to perform such that he is able to develop and expand his capacity to work independently. This can be emphasised after a new concept is introduced, when students are given the opportunity to clarify their doubts, to view the concept from different perspectives and to apply the concept on more structured and direct questions. Time must be set aside within the classroom to all of these. It is not healthy to rush through concepts and focus on the application immediately, without ensuring that students understand how the concepts work. Concepts introduced should be well-linked to previous concepts or sub-concepts so that students appreciate that it is an extension, or a synergy of several sub-concepts, and not merely as another formula which they are required to memorise.