# Profound Understanding of Fundamental Mathematics

To be honest, I hadn’t heard of PUFM (profound understanding of mathematics) until my first workshop for the math’s elective. I couldn’t even tell you one of the principles let alone four. However, now that I have done reading into Liping Ma’s research, it all makes sense. PUFM is essential if we as teachers are to enable learners to really get a grasp on mathematics.

So what is PUFM and why is it important? PUFM is more than just being able to understand elementary mathematics, it is being aware of the theoretical structure and attitudes of mathematics that are the foundations in elementary mathematics. Furthermore, being able to provide this foundation to instill those attitudes into learners. Having a PUFM is essential when teaching as in order for a learner to understand a concept, the teacher must fully understand it first both conceptually and procedurally.

Elementary mathematics is a range of depth, breadth and thoroughness. Teachers who are successful in achieving this deep, vast and thorough understanding will be able to reveal and represent the connections that lie between and among mathematical ideas in their teaching. These 4 interrelated properties are the principles that lead to these different aspects of meaningful understanding in mathematics:

Connectedness refers to connections made among mathematical concepts and procedures, anything from basic connections between individual strands of knowledge to more complex, underlying connections that link mathematical operations. When put into practice, this should ensure the learning that takes place doesn’t become fragmented. For example, instead of learning isolated topics, learners will engage with a unified body of knowledge.

Multiple Perspectives refers to when a teacher has achieved the PUFM and can appreciate different facets to an idea or various approaches to a solution including the advantages and disadvantage. Additionally, they can provide mathematical explanations in relation to these facets and approaches. In teaching, this can lead to learners having a flexible understanding of a concept.

Basic Ideas refers to displaying mathematical attitudes and being aware of the “simple but powerful basic concepts and principles of mathematics”. Teachers with PUFM will tend to revisit and reinforce these basic ideas. This leads students to be guided to conduct real mathematical activity.

Longitudinal Coherence refers to not limiting the knowledge that is being taught as you will have achieved a fundamental understanding of the whole curriculum. Therefore, you are ready to exploit any opportunity in order to review crucial concepts. Additionally, they have an understanding of what students will go on to learn later and take these opportunities to lay a real foundation.

All four of these principles are equally as important when gaining a sound understanding of elementary mathematics. This is extremely important when teaching mathematics as how can you teach and inspire young minds without having a deep understanding yourself? Not only is this relevant to maths but it is something I will keep in mind when teaching any subject as each of these principles can be applied to different topics. I found Liping MA’s research extremely eye opening and I will continue to use these principles and keep them in mind when bringing new mathematical concepts into my future classroom.