Profound Understanding of Fundamental Mathematics (PUFM)

The fundamental factors of mathematics are considered as the being the basic ideas in which a learner should have an understanding of in the first instance before progressing on to attempt more complex mathematical problems and processes.  In essence, this is knowledge is used as the foundations of which knowledge can be built upon thus enabling learning and understanding to evolve.  Liping Ma (2010) suggests that we must develop ‘a profound understanding of fundamental mathematics’ in order to be able to promote effective mathematical learning and in turn, teach it effectively.  This involves having a concrete knowledge of the structures embedded in mathematics and how they are used.

Primary school teachers require a certain level of mathematical knowledge to ensure that effective teaching is taking place.  Liping Ma (2010) identifies four key elements which all contribute to a person gaining a profound understanding of mathematics.  These four elements are ‘Connectedness’ which is the ability to be able to relate topics to others thus enabling the existing foundations of knowledge and understanding to be built upon allowing new concepts and operations to be processed.  ‘Multiple perspectives’ which can be described as the ability to vary the approach used in solving mathematical problems in which Ma (2010) suggests that if a person is successful in managing this process then their knowledge and understanding of that particular topic is considered complete. The next element is ‘Basic ideas’ being the ability to be able to identify the basic concepts and attitudes of mathematics and utilise these when informing future pathways in mathematical problem solving.  ‘Longitudinal coherence’ relates to the content learned from the beginning of the understanding of mathematical processes and how it influences the current understanding, regardless of how limited this may or may not be.  This concept is essential to teachers as they are essentially influencing the learners understanding from the start of learning and onwards through their learning journey.  Ma (2010) supports that teachers ‘are able to provide mathematical explanations of approaches’ and ‘can lead their students to a flexible understanding of the discipline’ (pg.122).  She also states that these four key elements ‘are the kind of connections that lead to different aspects of meaningful understanding of mathematics’ which is essential to teaching.  Teachers in possession of a vast and thorough understanding of mathematical concepts are able to constitute it through different rules and concepts.

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