Nature of Mathematics

The nature of mathematics has been the focus of much writing over the last few

decades (e.g., Begg, 1994, 2005; Dossey, 1992; Fuson, Kalchman & Bransford, 2005;Ocean, 2005; Presmeg, 2002; Winter, 2001). Dossey (1992) argues that differentconceptions of mathematics influence the ways in which society views mathematics. This can influence the teaching of mathematics, and communicate subtle messages to children about the nature of mathematics that “affect the way they grow to view mathematics and its role in their world” (p. 42). Similarly, Presmeg (2002) has argued that beliefs about the nature of mathematics either enable or constrain “the bridging process between everyday practices and school mathematics” (p. 295).

Many dichotomies exist that highlight the contrasting ways in which Mathematics is viewed.

Some hold an external conception of mathematics and believe that Mathematics is a fixed body of knowledge that is presented to students. On the other hand those favouring the internal conception believe that Mathematics is personally constructed and has individual meaning.

Some view Mathematics as Mathematical content including Knowledge and procedures. Others view it as Mathematical processes including reasoning, problem-solving, communicating and making connections.

There is also a tension between the mechanistic view of mathematics as in the development of skills and knowledge and Mathematics as a way of fostering citizenship and responsibility within society as in the development of personal, spiritual, moral, social, and cultural dimensions.

A distinction has been made between mathematical activity carried out for its own

sake, and mathematical activity that is useful for something else (Huckstep, 2000).

In order to distinguish between the aims and purposes of mathematics education, we as maths mentors need to ask a very important question

·         “What are we trying to do in mathematics education?”

·         “What are we trying to do it for?”.

This is also related to the debate about what is Mathematics and What is numeracy?

Definitions of numeracy emphasize the practical or everyday uses of mathematics in contexts such as homes, workplaces, and communities. People who argue that mathematics is valuable for its own sake often write about the beauty and aesthetics of mathematics, and the sheer enjoyment of doing mathematics.

 I personally feel that it is impossible to choose and favour one argument or the other in case of the four dichotomies mentioned.

1.                   I believe Mathematics is a fixed body of knowledge that we present to the students day after day year after year because as teachers we are away from the mathematical research that goes everyday . Neither are we aware of any new developments in the Mathematical body of knowledge nor do we keep ourselves equipped with the pedagogical changes that take place. So for us and our students Mathematics is essentially a fixed body of knowledge. The internal conception that students or teachers do can be seen in the way they comprehend a concept or a mathematical proble. These internal conceptions appear as either misconceptions and myths or as different ways and analogies for understanding Mathematics. For example Mathematics is internally constructed for me because I believe and teach that 0 is not a fraction but a rational number. I have my own researched reasoning for the same. I also construct personal meaning when I try to express a maths concept as a story or a play or a song or an activity.

2.                   Mathematics as Mathematical content including Knowledge and procedures forms the backbone of all our teaching planning, execution in class as well as assessment. But Mathematical processes including reasoning, problem-solving, communicating and making connections are also not left behind. I believe Knowledge and procedures are what you need to master first and then you can work at the higher cognitive level of developing and addressing reasoning, problem-solving and other such skills. I try to balance the two by presenting the mathematical knowledge and procedures in a way that my students are able to develop the higher skills as well. I try to let them go deep and ‘feel’ Mathematics so that all of their senses could be involved in learning the mathematical knowledge and processes. I use questioning to help them develop higher order thinking skills like reasoning – logical and critical. For problem Solving, I plan small real life simulated projects wherein they frame the problem, think of the possible solutions and test retest which is the most suitable strategy. Communication and connections are established in my maths classrooms by encouraging verbal presentations as individuals and groups.

3.                   For the mechanistic vs. holistic view of Mathematics, I again feel you need to master the mechanics of Mathematics first before mathematics could make you a global citizenship.

4.                   To the question ‘What is Mathematics’ , I have developed my individual favour towards the logic part of mathematics in comparison to the numeric part. Guiding factor – I have always been poor in numeracy. I don’t have a charm for numbers and their arithmetic. I don’t have that part of my brain functional that lets you manipulate numbers. Plain and clearly constructed personal reasoning!

There is no right or wrong argument as far as the dichotomies regarding the nature of mathematics are concerned. But it is important that we give a thought to all the views and all the perspectives. Because it is our view of the nature of mathematics that guides our philosophy and psychology of teaching Mathematics.