Magazine article Mathematics Teaching

Concept-Based Learning in Mathematics: Teaching for Deep Understanding in Secondary Schools

Magazine article Mathematics Teaching

Concept-Based Learning in Mathematics: Teaching for Deep Understanding in Secondary Schools

Article excerpt

More than one million primary and secondary students around the world pursue their studies by following the International Baccalaureate (IB) programmes. An increasing number of international schools are adopting IB programmes which have a central aim of developing inquiring, knowledgeable and caring young people who are motivated to succeed. IB programmes are respected for their academic rigour and for their emphasis on personal development. IB curricula for different discipline areas are designed drawing on the latest educational research and represent forward thinking and dynamic sets of standards. Our school adopted the IB programme in 2007 responding to the international nature of our students and because of the emphasis on academic challenge alongside personal development. In this article, I will discuss the new initiatives, drawn from the IB programme, which promote a coherent framework for the teaching and learning of mathematics. I will focus on teaching to promote conceptual understanding and draw on examples of how I try to promote deep understanding in the mathematics classroom.

In 2014, the IB released the Approaches to Teaching and Learning in the Diploma Programme. This outlines the pedagogical principles to be adopted across all IB programmes. One of the six pedagogical principles is that:

Teaching is focused on conceptual understanding.

The IB Middle Years programme, aimed at 11-14 year olds, has specifically adopted a concept-based curriculum model in mathematics using concepts such as form, relationships and logic. Concept-based curriculum and instruction was introduced by Dr H. Lynn Erickson in her book Concept-based Curriculum and Instruction for the Thinking Classroom (2007). This concept-based curriculum is a three-dimensional educational design model that frames facts and skills with a third layer, the conceptual layer. The concept-based model represents an approach to teaching and learning that focuses on creating a synergistic relationship between facts, skills and conceptual understandings in order to encourage thinking and to develop intellect. This relationship is illustrated below.

The hands in the graphic represent the synergistic relationship between facts, skills and conceptual understanding. This is achieved through the inquiry process continuum and is facilitated by inquiry based learning. Inquiry based learning is a student-centred approach that develops independence, curiosity and allows learners to construct their own meaning and results in a deeper understanding. I would also suggest that by drawing attention to concepts we are able to develop critical thinking, reasoning and deeper understanding of the content.

If we are to think about conceptual understanding we need to share a definition of a concept.

I think that concepts could be described as being universal organising ideas or mental constructs that share common attributes and that transcend culture and time. Examples of concepts within the topic of calculus would be rates of change and gradients.

Concept-based learning

In the school in which I work we are developing concept-based learning to include the following elements. At the start of the planning process we identify what students need to know, understand and do (we call this identifying the KUDs). An example of a KUD is given in table 1.

Concept-based learning focuses on generalisations, which can be seen as statements of conceptual understanding. These statements are sometimes referred to as enduring understandings and form the 'understanding1 part in KUDs. An example of such a statement for differentiation could be:

Tangents identify the instantaneous velocity or acceleration at a particular point in time.

Learners are supported in developing generalisations through the use of guiding questions, which are used to draw generalisations from our students in a unit of study. Guiding questions come in three forms. …

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