When I first started this module and we were introduced to the idea of ‘profound understanding of fundamental mathematics’ (PUFM), I was slightly terrified. It sounds terribly confusing but actually it boils down to very simple concepts. “By profound understanding I mean an understanding of the terrain of fundamental mathematics that is deep, broad and thorough” (Ma, 2010).
Connectedness – “A teacher with PUFM has a general intention to make connections among mathematical concepts and procedures.” (Ma, 2010). This simply means being able to identify ways in which mathematical concepts procedures connect to one another, and highlighting this when we teach so that children can then identify these links. In practice this would mean the learning of a child was not fragmented.
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). This means that the teacher respects the different aspects of problems and solutions and allows children to explore these different aspects in order that they have a flexible understanding of the subject.
Basic Ideas – “Teachers with PUFM display mathematical attitudes and are particularly aware of the “Simple but powerful basic concepts and principles of mathematics” (e.g. the idea of an equation)” (Ma, 2010). This means that teachers encourage children to explore the ideas in relation to a problem as opposed to simply calculating the solution. This will mean their learning and understanding of the subject will be more in depth.
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). This means that teachers are willing to revisit learning done in previous years, but also to plan accordingly with the flow of the classrooms curriculum and meet the present needs of the child within their studies.
PUFM is more than simply understanding the subject area that you are teaching. It is embracing the subject as a whole and appreciating the foundations of the subject. Having a profound understanding is what we aspire to as teachers. If we are expectant of a child to have great depth of a subject, then so must we.
Liping Ma, 2010, Knowing and Teaching Elementary Mathematics, New York, Routledge, p.104
