Profound Understanding of Fundamental Mathematics (PUFM)

When I was first asked to describe what profound understanding of fundamental mathematics (PUFM) meant, I immediately thought it meant being a maths genius. It sounded very complicated. However it soon become clear that this was not true.

Liping Ma created four universal principles that would enable a teacher to have a profound understanding of fundamental mathematics. These four principles consisted of inter-connectedness, multiple perspectives, basic ideas and longitudinal coherence. Ma (2010) believes if a teacher can demonstrate these four principles within a classroom then they have a profound understanding of fundamental mathematics.

 Inter Connectedness can be describedas the ability to make connections between different concepts. When teachers apply this principle within the classroom pupils learning will become one body of knowledge instead of being disjointed.

Multiple perspectivesis the idea of seeing and appreciating different approaches to solving a problem. This stops your pupils being restricted to a specific learning cycle as there are multiple ways to reach the same answer. Demonstrating that the myth “there is only one process of finding an answer” is incorrect. It allows a more flexible way of thinking within mathematics.

 Basic concepts within mathematics refers to the basic ideas such as ordering. We must be aware of the basic ideas and apricate their importance. Successful teacher will revisit and reinforce these basic ideas throughout their pupils learning, as they provide the foundation upon which future concepts are learned.

Longitudinal coherence is an understanding of the whole curriculum and how one idea can build on another idea. A teacher with PUFM can identify where a pupil is with their learning and support them with the correct learning to build their mathematical knowledge and skills. They are also aware of the different mathematical concepts their class is going to cover in the future, allowing them to effectively lay solid foundations ahead of time.

After looking into the four inter-related principles which make up mathematics I can see the importance of learning mathematics and exploring the different aspects of mathematics around us. When I become a qualified teacher I will highlight all four principles within my classroom to allow the best learning and teaching for my pupils.

Moving forward in this module I would like to study the wider context of mathematics and how mathematics is used in everyday life through play, music and art. I also want to explore the different ways of making mathematics meaningful and exciting for the pupils as I understand the importance of inspiring young children to want to learn mathematics for themselves and not because they have to.


Ma, L. (2010). Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States. 2nd ed. New York: Routledge.


1 thought on “Profound Understanding of Fundamental Mathematics (PUFM)

  1. Jonathan Brown

    I will be interested to read your blog posts about some of these areas. Remember to set yourself a specific question that you want to answer or investigate as this will help you to engage in a discussion. You might also find it useful to have a look at the other blogs from this elective.


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