During our first input, we were introduced to Liping Ma’s concept of profound understanding of fundamental mathematics (PUFM) and its importance in terms of enhancing teachers’ knowledge of, and ability to better teach, elementary mathematics.
Ma states that to “fully promote mathematics learning, teachers must first have a profound understanding of fundamental mathematics”. But having a PUFM is more than being able to understand elementary mathematics, it’s about being aware of the theoretical structure and basic attitudes of mathematics that are within elementary mathematics, and being able to use this as a foundation for which to instil the same awareness in the children that we teach. It’s important to have both a conceptual and procedural understanding – to know how and why we do something – that is deep, broad and thorough.
Ma identified four principles that are essential to gaining PUFM:
(Inter) connectedness: refers to being able to see connections between concepts and procedures. These connections allow for the use of prior learning and knowledge to be applied to new mathematical situations. This helps to ensure learning is not fragmented, but viewed instead as a unified body of knowledge.
Multiple perspectives: refers to seeing and appreciating different approaches to solving a problem. This encourages a much more flexible way of thinking as it is not restricted to any specific learning style.
Basic concepts: refers to being aware of the basic concepts within mathematics. It is important that these central ideas are revisited and reinforced as they provide the foundation upon which future concepts are learned.
Longitudinal coherence: refers to being aware of the entire mathematics curriculum and how this can be used to link previously learned knowledge with newly acquired knowledge (one basic idea/concept builds on another). This allows for there to be much more understanding and flexibility in terms of where learning is headed as lessons can be tailored with this in mind.
While I’m still getting to grips with what PUFM really means for me as a teacher, it’s amazing how being made aware of these four inter-related principles has really changed the way I look at mathematics, and will definitely influence how I teach mathematics in the future.
References
Ma, L. (2010) Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States. 2nd edition. New York: Routledge.
An interview with Liping Ma: https://www.eduplace.com/intervention/knowingmath/liping_ma/conversation.html
