If someone asked me two months ago what ‘profound understanding of fundamental mathematics’ was, all I could tell you it was something to do with maths. PUFM sounds very confusing and complicated, when in fact it is quite the opposite. Over the course of this module, my understanding of this statement has developed greatly, and I now feel much more confident in my abilities of teaching in a classroom setting.

Ma (2010) defines PUFM as “an understanding of the terrain of fundamental mathematics that Is deep, broad and thorough.” There are four principles in teaching and learning that represent a teachers understanding of fundamental maths in the classroom: **interconnectedness, multiple perspectives, basic ideas and longitudinal coherence.**

**Interconnectedness – **

Ma (2010) defines interconnectedness as when “a teacher with PUFM has a general intention to make connections among mathematical concepts and procedures.” Interconnectedness occurs when the learner is trying to make a connection between the mathematical concepts and procedures. Here, it is the teachers job to stop the learning from being fragmented, and help the students to develop the ability to make links between underlying mathematical concepts, and showing how maths topics rely on each other.

**Multiple Perspectives –**

“Those who have achieved PUFM appreciate different facets of an idea and various approaches to a solution, as well as their advantages and disadvantages” (Ma, 2010). Multiple perspectives is when a learner can approach mathematical problems in many ways, as they understand the various perspectives when taking on a maths questions, and are able to look at the pros and cons of the different viewpoints. It is the teachers job to provide opportunities for the learners to be able to think flexibly about the thinking and understanding of different concepts in maths.

**Basic Ideas –**

MA (2010) says that “teachers with PUFM display mathematical attitudes and are particularly aware of the ‘Simple but powerful basic concepts of mathematics’ (e.g. the idea of an equation.” Basic ideas is a way of thinking about maths in terms of equations. Ma (2010) has suggested that learners should be guided to conduct ‘real’ maths activities rather than just approaching the problem when practising the property of basic ideas. If a teacher is implementing this principle effectively, then they will not only be motivating the learner to approach the maths problems, but will also be providing a guide to for the learners to help them understand the maths themselves.

**Longitudinal Coherence –**

“Teachers with PUFM are not limited to the knowledge that should be taught in a certain grade; rather they have achieved a fundamental understanding of the whole elementary mathematics curriculum” (Ma, 2010). Longitudinal coherence is when a learner recognises that each basic idea builds on each other. When a learner does not have a limit of knowledge, it is impossible to identify the level or stage that a learner is working at in maths. the learner has instead achieved a holistic understanding of maths (fundamental). A teacher who has a profound understanding of maths is one who can identify the learning that has previously been obtained. The teacher will then lay the fundamental maths as a foundation for learning later on (Ma, 2010).

The four principles are vital to having a deep understanding of maths, especially when teaching the subject. I will make sure I have PUFM before I begin teaching mathematics, as without it, I cannot teach the subject as in depth and as thoroughly as possible.

*References:*

*Ma, L. (2010). Knowing and Teaching Elementary Mathematics. New York: Routledge.*