Liping Ma (2010) suggests that to have a profound understanding of fundamental mathematics (PUFM) that you must have these 4 main principles which are, basic ideas, connectedness, multiple perspectives and longitudinal coherence. In this post, I will discuss my understanding of these 4 principles.

__Basic Ideas__

Liping Ma (2010, p.122) states that:

*“ Basic Ideas. Teachers with PUFM display mathematical attitudes are particularly aware of the “simple but powerful concepts and principles of mathematics” (e.g. the idea of an equation). They tend to revisit and reinforce these basic ideas. By focusing on these basic ideas, students are not merely encouraged to approach problems, but are guided to conduct real mathematical activity.”*

Children should be taught basic ideas in maths in order to progress this idea into a more complex understanding of mathematics. Basic ideas should be easy to grasp before making the next step to further their understanding of a mathematical idea. If children are able to understand the basic idea in maths, it is important that this is reinfored before moving on to something more complicated.

If we start with the basic idea of addition and subtraction. It is important that before moving on to a topic such as money, they must have grasped how to add and subtract or they will be unable to understand how to use money correctly.

__Connectedness__

Connectedness in mathematics is a concept within maths that is connected to several other ideas.

*“ Connectedness. A teacher with PUFM has a general intention to make connections among mathematical concepts and procedures, from simple and superficial connections between individual pieces of knowledge to complicated and underlying connections among different mathematical operations and subdomains. When reflected in teaching, this intention will prevent students’ learning from the being fragmented. Instead of learning isolated topics, students will learn a unified body of knowledge.”* (Liping Ma 2010, p.122)

This means that for a teacher to achieve connectedness they should not just teach maths as a single topic but link it to other topics which is an important factor as children need to be made aware as to why they are learning the mathematics in the first place. Children should be taught to make connections between mathematics and the real world and how maths is connected in everyday life, even if they don’t initially see it. For example, children can connect maths to things that they do in their everyday lives such as, baking and even going to the shops.

When teaching maths to children, it would be useful to incorporate this into different lessons. Instead of just teaching a maths lesson, a teacher could do a baking lesson and incorporating maths to highlight that these are connected. When teaching about money, children will be able to relate to this if they have been shopping with their parents. Looking at price labels, calculating the percentage discounts of reduced items and handling money and calculating change at the checkout can all contribute to children’s mathematical learning.

This connection to everyday life is important for children to understand as this will make their learning in class more meaningful to them.

__Multiple Perspectives__

Liping Ma (2010, p.122) states that to have a profound understanding of fundamental mathematics you must have multiple perspectives.

*“ 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. In addition, they are able to provide mathematical explanations of these various facets and approaches. In this way, teachers can lead their students to a flexible understanding of the discipline.”*

This means that, when teaching maths in school, a teacher should have several different ways of coming up with a solution rather than one specific way. Sometimes when looking at something in a different perspective you are able to find the answer if you are struggling to do so. By providing multiple perspectives in class, it makes mathematics feel less restrictive to children. It is important that children should be taught to have multiple perspectives within mathematics as they will be able to see that there may be other ways to solve a problem, which may make solving a problem easier for them.

__Longitudinal Coherence__

Ma (2010, p.122) finally states that to have a profound understanding of mathematics, one must have longitudinal coherence.

*“ 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. With PUFM, teachers are ready at any time to exploit an opportunity to review crucial concepts that students have studied previously. They also know what students are going to learn later, and take opportunities to lay the proper foundation for it.”*

This principle is one that I had great difficulty understanding at first. This suggests that when learning maths, the learner does not have a limit of how much knowledge that they have. I read into longitudinal coherence more and came across some reading that helped explain this so that even I could understand. Wu (2002) made this a lot easier for me to comprehend. Wu suggests that, as teachers, we make children of “individual trees clearly but, in the process, we short-change them by not calling their attention to the forest.” Which suggests to me that when teaching mathematics, we may only point out the most basic ideas but not the whole complex idea itself.

Wu (2002) also states that:

“We produce many students who do not think globally — or to use a more common word these days, holistically — about mathematics. In the present context, teachers who come through such a program may know the individual pieces of the school curriculum, but they are less adept at seeing the interrelationships among topics of different grades.”

I feel that Wu (2002) provided me with a far less complicated idea when it comes to longitudinal coherence. When teaching, it is important that encourage children to understand mathematics when teaching new topics. Pupils must be able to draw on their previous knowledge in mathematics to help them to understand new concepts. I feel that this is important to continue this throughout the children’s school life. Children should be able to draw on their previous learning in order to help them with their future development.

Although this concept still confuses me, I feel that I have developed a better understanding on what longitudinal coherence is.

Liping (2010) Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in china and the United States. 2nd edn. New York: Taylor & Francis.

Wu, H. (2002) What is so difficult about the preparation of mathematics teachers? Available at: https://math.berkeley.edu/~wu/pspd3d.pdf (Accessed: 31 October 2016).