Nature of Mathematics
1. Introduction
2.1 What is mathematics all about? The assignment brief suggests two viewpoints: (1) Mathematics is a given body of knowledge and standard procedures that has to be covered or (2) Mathematics is an interconnected body of ideas and reasoning processes
2.2 The first viewpoint considers mathematics as a discipline consisting of rigid compartments of knowledge with set techniques and routine algorithms. The second viewpoint suggests that mathematics is made up of interlinking ideas to be developed through experimenting and investigation.
2.3 From a teaching and learning point of view, one’s conception of the nature of mathematics is considered to have a profound impact on one’s teaching practice. According to Hersh (1986), the issue is not “What is the best way to teach but, What is mathematics really all about?” (Grouws, 1992, page 127).
2.4 The perception of the nature of mathematics not only influences how the subject is taught, but also has implications on how mathematics education for school is defined. Indeed, Ernest (1991) states that the view of the nature of mathematics together with the social and political contexts are seen as the key factors affecting curriculum planning (Ernest, 1991, page 125).
2. Literature Review
3.5 Lerman (1990) has identified two contrasting themes concerning the nature of mathematics, namely the absolutist and fallibilist views. According to him, they correspond to the two competing schools of thought in the philosophy of mathematics suggested by Lakatos (1978): Euclidean and Quasi-empirical. The former considers mathematics to be based on a body of unchallengeable truths whilst the latter sees mathematical knowledge a result of a discovery process involving conjectures, hypotheses and refutations. 3.6 Ernest (1996) explained that absolutism considers mathematics to be timeless and a priori knowledge. In his view, teachers who hold such a conception of mathematics would present the subject as a collection of unrelated facts and rules to be studied (Ernest, 1996).
3.7 In contrast, a fallibist considers mathematics a dynamic and an evolving field of study and its knowledge open to refutations and revisions. According to Tymoczko (1986), the fallibilist does not consider mathematics to be a body of pure and abstract knowledge, unconnected to social practices and ahistorical (Ernest, 1996).
3.8 How then do the modern mathematicians view mathematics themselves? Hersh (1986) argues that “Mathematics deals with ideas. Not pencil marks or chalk marks…” (Tymoczko, 1998, page 22). He explains that mathematical knowledge comes about from investigation. For example:
To have the idea of counting, one needs the experience of handling coins or blocks or pebbles. To have the idea of an angle, one needs the experience of drawing straight lines that cross… (Tymoczko, 1998, page 24)
3.9 This aligns well with the fallibilist way of thinking as it suggests that mathematical knowledge is gained through exploration and proofs as this is when ideas are developed, and not from reproducing a method or procedure demonstrated by the teacher.
3.10 My initial view of the nature of mathematics was an absolutist one in that I saw mathematics as a body of knowledge consisting of formulae and algorithms to be learned, memorised and applied. This was how I learned mathematics at school.
3.11 Using trigonometry as an example, I now have a tangible understanding of sine and cosine from exploring the coordinates of the end point of a rotating arm (Hibernia College, 2012, SKE Module 1: Square Numbers and Circular Measures). I now see cosine as the x-coordinates of the end point of a unit length arm rotated by an angle and sine the y-coordinates. Whereas in school, I was taught that cosine and sine are ratios between two of the three dimensions of a right-angle triangle. This latter method effectively undercuts the base on which trigonometry was built upon and ignores the process from which it was derived. The former approach on the other hand demystifies trigonometry and provides a physical meaning to what it actually represents.
3.12 My view of the nature of mathematics now tends towards fallibilism. I no longer see mathematics as an abstract study. Instead, I see it as a body of ideas that have been developed through a cycle of exploration and proof.
3. Conclusion 4.13 Real mathematicians gain new mathematical knowledge through investigation. They perceive mathematics as a body of ideas to be experimented, tinkered with and verified. They do not consider mathematics to be a body of knowledge with a set of standard procedures that cannot be physically represented and visualised. In my opinion, the learning of mathematics should not be any different.
4.14 My experience of re-learning mathematics has revealed to me that mathematics is best learned through investigation. For example, in my learning of basic Trigonometry I have developed a more in-depth understanding of sine and cosine and I no longer see trigonometry as a field of abstract study that is difficult to explain. I consider this in-depth knowledge to be paramount to teaching as without which, it would be difficult to provide an insightful view to the pupils.
4.15 According to the QTS standards, teachers need to “have a secure knowledge of the relevant subject(s) and curriculum areas, foster and maintain pupils’ interest in the subject, and address misunderstandings”. In my opinion, it would be difficult to meet this standard if teachers do not move away from presenting mathematics as a body of facts and set procedures to be learned and memorised. Research has shown that this narrow view of teaching mathematics is more likely to result in pupils developing shallow knowledge and very often a resentment towards the subject (Johnston-Wilder et. al., page 209).
REFERENCES
DfEE (1999) Mathematics: the National Curriculum in England. London: HMSO.
Ernest, P. (1991). The Philosophy of Maths Education London: The Falmer Press.
Ernest, P. (1996). The Nature of Mathematics and Teaching. In Philosophy of Mathematics Education Newsletter 9, POME. University of Exeter, UK. See http://people.exeter.ac.uk/PErnest/pome/pompart7.htm.
Grouws, D. (1992). Handbook of Research on Mathematics Teaching and Learning New York: Macmillan.
Hibernia College (2012) Subject Knowledge Enhancement Module 1: Square Numbers and Circular Measures.
Johnston-Wilder, S. et. al. (2011). Learning to Teach Mathematics in the Secondary School London and New York: Routledge.
Lerman,S. (1990). Alternative Perspectives of the Nature of Mathematics. British Educational Research Journal, Vol. 16, Issue 1.
QCA(2007) National Curriculum. London: Qualifications and Curriculum Authority. See http://curriculum.qcda.gov.uk/key-stages-3-and-4/index.aspx.
Tymoczko, T. (Ed.). (1998). New directions in the philosophy of mathematics. Boston: Birkhauser.
References: DfEE (1999) Mathematics: the National Curriculum in England. London: HMSO. Ernest, P. (1991). The Philosophy of Maths Education London: The Falmer Press. Ernest, P. (1996). The Nature of Mathematics and Teaching. In Philosophy of Mathematics Education Newsletter 9, POME. University of Exeter, UK. See http://people.exeter.ac.uk/PErnest/pome/pompart7.htm. Grouws, D. (1992). Handbook of Research on Mathematics Teaching and Learning New York: Macmillan. Hibernia College (2012) Subject Knowledge Enhancement Module 1: Square Numbers and Circular Measures. Johnston-Wilder, S. et. al. (2011). Learning to Teach Mathematics in the Secondary School London and New York: Routledge. Lerman,S. (1990). Alternative Perspectives of the Nature of Mathematics. British Educational Research Journal, Vol. 16, Issue 1. QCA(2007) National Curriculum. London: Qualifications and Curriculum Authority. See http://curriculum.qcda.gov.uk/key-stages-3-and-4/index.aspx. Tymoczko, T. (Ed.). (1998). New directions in the philosophy of mathematics. Boston: Birkhauser.
2.1 What is mathematics all about? The assignment brief suggests two viewpoints: (1) Mathematics is a given body of knowledge and standard procedures that has to be covered or (2) Mathematics is an interconnected body of ideas and reasoning processes
2.2 The first viewpoint considers mathematics as a discipline consisting of rigid compartments of knowledge with set techniques and routine algorithms. The second viewpoint suggests that mathematics is made up of interlinking ideas to be developed through experimenting and investigation.
2.3 From a teaching and learning point of view, one’s conception of the nature of mathematics is considered to have a profound impact on one’s teaching practice. According to Hersh (1986), the issue is not “What is the best way to teach but, What is mathematics really all about?” (Grouws, 1992, page 127).
2.4 The perception of the nature of mathematics not only influences how the subject is taught, but also has implications on how mathematics education for school is defined. Indeed, Ernest (1991) states that the view of the nature of mathematics together with the social and political contexts are seen as the key factors affecting curriculum planning (Ernest, 1991, page 125).
2. Literature Review
3.5 Lerman (1990) has identified two contrasting themes concerning the nature of mathematics, namely the absolutist and fallibilist views. According to him, they correspond to the two competing schools of thought in the philosophy of mathematics suggested by Lakatos (1978): Euclidean and Quasi-empirical. The former considers mathematics to be based on a body of unchallengeable truths whilst the latter sees mathematical knowledge a result of a discovery process involving conjectures, hypotheses and refutations. 3.6 Ernest (1996) explained that absolutism considers mathematics to be timeless and a priori knowledge. In his view, teachers who hold such a conception of mathematics would present the subject as a collection of unrelated facts and rules to be studied (Ernest, 1996).
3.7 In contrast, a fallibist considers mathematics a dynamic and an evolving field of study and its knowledge open to refutations and revisions. According to Tymoczko (1986), the fallibilist does not consider mathematics to be a body of pure and abstract knowledge, unconnected to social practices and ahistorical (Ernest, 1996).
3.8 How then do the modern mathematicians view mathematics themselves? Hersh (1986) argues that “Mathematics deals with ideas. Not pencil marks or chalk marks…” (Tymoczko, 1998, page 22). He explains that mathematical knowledge comes about from investigation. For example:
To have the idea of counting, one needs the experience of handling coins or blocks or pebbles. To have the idea of an angle, one needs the experience of drawing straight lines that cross… (Tymoczko, 1998, page 24)
3.9 This aligns well with the fallibilist way of thinking as it suggests that mathematical knowledge is gained through exploration and proofs as this is when ideas are developed, and not from reproducing a method or procedure demonstrated by the teacher.
3.10 My initial view of the nature of mathematics was an absolutist one in that I saw mathematics as a body of knowledge consisting of formulae and algorithms to be learned, memorised and applied. This was how I learned mathematics at school.
3.11 Using trigonometry as an example, I now have a tangible understanding of sine and cosine from exploring the coordinates of the end point of a rotating arm (Hibernia College, 2012, SKE Module 1: Square Numbers and Circular Measures). I now see cosine as the x-coordinates of the end point of a unit length arm rotated by an angle and sine the y-coordinates. Whereas in school, I was taught that cosine and sine are ratios between two of the three dimensions of a right-angle triangle. This latter method effectively undercuts the base on which trigonometry was built upon and ignores the process from which it was derived. The former approach on the other hand demystifies trigonometry and provides a physical meaning to what it actually represents.
3.12 My view of the nature of mathematics now tends towards fallibilism. I no longer see mathematics as an abstract study. Instead, I see it as a body of ideas that have been developed through a cycle of exploration and proof.
3. Conclusion 4.13 Real mathematicians gain new mathematical knowledge through investigation. They perceive mathematics as a body of ideas to be experimented, tinkered with and verified. They do not consider mathematics to be a body of knowledge with a set of standard procedures that cannot be physically represented and visualised. In my opinion, the learning of mathematics should not be any different.
4.14 My experience of re-learning mathematics has revealed to me that mathematics is best learned through investigation. For example, in my learning of basic Trigonometry I have developed a more in-depth understanding of sine and cosine and I no longer see trigonometry as a field of abstract study that is difficult to explain. I consider this in-depth knowledge to be paramount to teaching as without which, it would be difficult to provide an insightful view to the pupils.
4.15 According to the QTS standards, teachers need to “have a secure knowledge of the relevant subject(s) and curriculum areas, foster and maintain pupils’ interest in the subject, and address misunderstandings”. In my opinion, it would be difficult to meet this standard if teachers do not move away from presenting mathematics as a body of facts and set procedures to be learned and memorised. Research has shown that this narrow view of teaching mathematics is more likely to result in pupils developing shallow knowledge and very often a resentment towards the subject (Johnston-Wilder et. al., page 209).
REFERENCES
DfEE (1999) Mathematics: the National Curriculum in England. London: HMSO.
Ernest, P. (1991). The Philosophy of Maths Education London: The Falmer Press.
Ernest, P. (1996). The Nature of Mathematics and Teaching. In Philosophy of Mathematics Education Newsletter 9, POME. University of Exeter, UK. See http://people.exeter.ac.uk/PErnest/pome/pompart7.htm.
Grouws, D. (1992). Handbook of Research on Mathematics Teaching and Learning New York: Macmillan.
Hibernia College (2012) Subject Knowledge Enhancement Module 1: Square Numbers and Circular Measures.
Johnston-Wilder, S. et. al. (2011). Learning to Teach Mathematics in the Secondary School London and New York: Routledge.
Lerman,S. (1990). Alternative Perspectives of the Nature of Mathematics. British Educational Research Journal, Vol. 16, Issue 1.
QCA(2007) National Curriculum. London: Qualifications and Curriculum Authority. See http://curriculum.qcda.gov.uk/key-stages-3-and-4/index.aspx.
Tymoczko, T. (Ed.). (1998). New directions in the philosophy of mathematics. Boston: Birkhauser.
References: DfEE (1999) Mathematics: the National Curriculum in England. London: HMSO. Ernest, P. (1991). The Philosophy of Maths Education London: The Falmer Press. Ernest, P. (1996). The Nature of Mathematics and Teaching. In Philosophy of Mathematics Education Newsletter 9, POME. University of Exeter, UK. See http://people.exeter.ac.uk/PErnest/pome/pompart7.htm. Grouws, D. (1992). Handbook of Research on Mathematics Teaching and Learning New York: Macmillan. Hibernia College (2012) Subject Knowledge Enhancement Module 1: Square Numbers and Circular Measures. Johnston-Wilder, S. et. al. (2011). Learning to Teach Mathematics in the Secondary School London and New York: Routledge. Lerman,S. (1990). Alternative Perspectives of the Nature of Mathematics. British Educational Research Journal, Vol. 16, Issue 1. QCA(2007) National Curriculum. London: Qualifications and Curriculum Authority. See http://curriculum.qcda.gov.uk/key-stages-3-and-4/index.aspx. Tymoczko, T. (Ed.). (1998). New directions in the philosophy of mathematics. Boston: Birkhauser.