Liping Ma (2010) suggested that in order for mathematics to be understood correctly teacher’s must have a profound understanding of mathematics (PUFM). Ma said that there were four key concepts that you needed to understand in order to have a profound understanding of mathematics. These concepts are basic ideas, multiple perspectives, longitudinal coherence and connectedness. i am going to discuss my understanding of these in this post.
Basic Ideas
Ma (2010, p.122) describes this as ” 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). 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.”
The basic ideas of mathematics are really important as children need these basic ideas to be able to continue to more complex areas. The basic ideas are similar to the foundations a building. If these basic ideas are not taught correctly to children or some areas are missed out that will make it more difficult for children to learn the more difficult areas of mathematics that are built on top of these basic ideas. These basic ideas need to be reinforced so that children have a sound understanding of them. For example, children need to learn addiction and subtraction before they can move on to more complicated areas like money and time.
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)
Connectedness is a mathematical concepts that means math is connected to other subjects. I have realised through this module how important this principle is as there is math in everything.
In order for teachers to achieve connectedness they need to teach mathematics beside the areas that it can relate to and not as a single subject. Since math is laced through nearly everything that we do, that is how it needs to be taught. For children to have a thorough understanding on mathematics it is best to teach it alongside the other subjects that it is connected to.
A teacher could teach a music lesson and whilst doing different values of musical notes could do some work on counting and addiction. Another example would be a baking lesson, weight and measurement could be taught while measuring the ingredients. This gives the children some context to their lessons rather than just weighing random items that they do not care much about.
Multiple Perspectives
Liping Ma (2010) discribes multiple perspectives as “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.”
Multiple perspectives means that children need to have several ways to get to the answering not just one. This is useful for children as not everybody’s brains work in the same way. This means that if a child does not understand the first way that a new mathematical area is taught then the second or third way might make more sense. By multiple perspectives being provided in the classroom, it makes mathematics less restrictive and easier to understand.
Longitudinal Coherence
Lastly, Ma (2010, p.122) states that longitudinal coherence is needed to have a profound understanding of mathematics.
Ma describes this as “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 concept is the one I found most difficult to understand. However, it is still important but just takes a little longer to understand. This principle says that what a child learns should not be limited because of their age and year that they are in. Ma says that a teacher should be able to teach any mathematics to suit the level of his/her children. I think that this is vitally important for in the classroom as children should not be limited just because they are a certain age. If a child understands all of the areas that they “should” know in their year then why should they not be moved on to the next area even though it is normally done in the next year up. This means that teachers need to be prepared to teach children of all levels.
References
Liping (2010) Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in china and the United States. 2nd edn. New York: Taylor & Francis.
