When I was first asked to describe what profound understanding of fundamental mathematics (PUFM) meant, the panic and fear immediately set in. To me this was something that seemed really complicated and confusing however upon further research and reading I have discovered this is not the case.
Liping Ma believes that there are four different properties in both teaching and learning which highlight a teacher’s profound understanding of fundamental mathematics that can be displayed within the classroom. These four properties include interconnectedness, multiple perspectives, basic ideas and longitudinal coherence. Therefore, Ma means that if a teacher can show these four properties successfully then they have a profound understanding of fundamental mathematics.
Interconnectedness is the idea that mathematic topics depend on one another. Ma (2010) describes a teacher with PUFM has making a great effort to show the connections between different mathematical concepts and procedures. She believes that when this is applied within a classroom then the pupil’s learning will not become fragmented and instead of being separate individual lessons, it will become one unified body of knowledge.
Multiple perspectives can be described as the ability to approach mathematical problems in many ways. This can be highlighted within maths as there are many different ways to reach the same answer and all of those different ways can be correct. Liping Ma (2010) says that teachers who have achieved PUFM are able to appreciate different ideas and approaches to a problem, taking into account the advantages and disadvantages. Also on top of this, teachers within a classroom should be able to explain the different mathematical concepts to a problem. By doing this, the teacher allows their pupils to develop a flexible understanding of mathematics and how it works.
The basic ideas within mathematics refer to the basic principles such as place value and ordering. Ma (2010) states that those with PUFM will show an awareness for these basic principles and appreciate their importance. These teachers will often revisit and reinforce these basic ideas and principles. This then means that pupils are not only encouraged to approach problems, but they are guided to carry out proper mathematical activity.
Longitudinal coherence can be explained as the way that one basic idea can build on another idea. Liping Ma (2010) believes that in order for a teacher to achieve PUFM then they should have a holistic view of the curriculum. These teachers are not just limited to the knowledge required for their stage as they have an understanding of the whole curriculum. Within a classroom, a teacher with PUFM is ready to take every opportunity to review crucial concepts and ideas that the pupils have covered previously. Longitudinal coherence can also be displayed within a classroom as the teacher will know what mathematical concepts that the class is going to cover in the future. They will prepare the pupils effectively by laying the proper foundations ahead of time.
These four properties are crucial to my understanding of mathematics and how to teach the subject. When I have my own class one day I will ensure I am a teacher that shows connectedness, highlights multiple approaches to problems, revisits and reinforces basic ideas and lays the proper foundations for future teaching.
Ma, L (2010). Knowing and Teaching Elementary Mathematics. Oxon: Routledge. p120-125.
