Before starting the “discovering maths” module, I had never heard of PUFM (profound understanding of fundamental maths). To be honest I thought it sounded quite daunting, however, after doing some research and breaking it down I discovered that PUFM is not quite as intimidating as I had thought. It consists of 4 principles and according to Lipping MA (2010), “to fully promote mathematical learning, teachers must have a profound understanding of fundamental mathematics”. In other words teachers must have a deep, broad and thorough understanding of the mathematical topics in which they are and will be teaching (Ma, 2010).

If you are still as confused as i was initially, think of a bus driver. Bus drivers know the roads well and know how to make short cuts and reroute if needs be. Teachers with PUFM are like bus drivers, only relating to maths. They know the topics they are teaching very well and can take students from their understanding of maths to future learning, and in the process of this they know many ways in which the journey can do it (National Research Council, 2001).

The four principles in which PUFM is made up of are:

• Interconnectedness: A teacher with PUFM is able to identify and make connections between the mathematical topics they are teaching (Ma, 2010). They are able to make links with the pupils current knowledge and their future learning, expressing how these depend on each other.
• Multiple Perspective: This refers to a teacher being able to approach solutions and mathematical problems in a number of ways (Cuarezma, 2013). This can ensure flexible learning within the classroom as teacher are able to adapt their teaching and give mathematical explanation in regards to their approach (National Research Council, 2001).
• Basic Ideas: This involves teachers knowing the, ‘simple but powerful basic concepts and principles of mathematics’. This allows teachers to be able to reinforce basic maths ideas within students, as some of the most basic principles in maths recur through most learning (Cuarezma, 2013).
• Longitudinal Coherence: As Teachers, we have to understand that one mathematical topic builds on another, and what we teach students today is a base for future learning (Ma, 2010). A teacher must be able to revisit topics to sole date learning but also move on with their teaching in order to cover the whole curriculum while ensuring the needs of the learners are being met (Curezma, 2013).

Having PUFM as a teacher is essential as teachers must have a deep and thorough understanding of  the concepts they are teaching in order to teach them to others. I will be remembering the 4 principles to PUFM when I teach maths in the future to identify how they can be applied when teaching each topic.

Cuarezma, A. (2013). Q & A with Liping Ma: The New York Times. (online). Available at: html http://www.nytimes.com/2013/12/18/opinion/q-a-with-liping-ma.html (Accessed 1 Nov. 2017).

Ma, L. (2010) Knowing and Teaching Elementary Maths. 2^nd ed. New York: Routledge Publications. pp, 12, 14-19, 34-37.

National Research Council. (2001) Knowing and Learning Mathematics for Teaching. 1^st ed. Washington. The National Academies Press. pp, 11-20.