Nature of Mathematics and Pedagogical practices

Nature of Mathematics and
Pedagogical Practices
Presented by:
Laxman Luitel
Mphil Scholar, Kathmandu University School of Education
Teacher, Aksharaa School, Kathmandu, Nepal
3/24/2019
10th International Conference on Mathematics Education and
Society, Hyderabad, India
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Major Highlights
• Introduction/setting the scene
• Purpose of the study
• Nature of mathematics
• Historical background of nature of mathematics
• Pedagogical practices in mathematics classroom
• Conclusion
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Setting the Scene
• Formally, mathematics has been teaching
from the very beginning of the school level
• With the skills of four fundamental
operations
• In my experience, it was interesting to learn
mathematics in lower level than in higher
level
• However, students’ and teachers’ perception
towards the nature of mathematics have great
influence in teaching and learning process
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Contd…
• Dossey (1992) mentioned that perception of the nature of
mathematics held by the society has a major influence during the
development of school mathematics curriculum, instruction and
research
• In the context of Nepal, most of the mathematics teachers have
been habitual to respond that; it is difficult, abstract, exact,
important for the future etc. to the different epistemological
question related to the mathematics raised by the students
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• Very less people have been talking about
the contextual mathematics, ethno-
mathematics, etc.
• However, students’ and teachers’ engaging
with mathematics depends on their
perception of the nature of mathematics.
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Contd…

Purpose
• The main purpose of writing this paper is to unpack my
understanding about the nature of mathematics based upon the
literature and my experience of learning and teaching
mathematics as well as its impact in teaching and learning
mathematics.
• Hope that writing this paper enables me as well as other
mathematics teacher educators to transform the beliefs towards
nature of mathematics from absolutists to falibilist as well as to
improve teaching practices
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Nature of Mathematics
• Nature of mathematics has been talking by different
mathematicians, researchers, teacher educators, etc.
• Thoughts or perceptions of the nature of mathematics might be
determined by their beliefs and understanding towards mathematics
• In my initial phase of teaching, how and what I learned in
mathematics is important than why I learned
• Used to reproduce the same culture in my classroom what I
experienced
• Thus, memorizing the formula, theorems, solution steps, etc. were
the dominant practices in my mathematics classroom
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Contd…
• I realized that my experience of learning mathematics (school &
bachelor) is guided by the absolutist nature (Ernest, 1991) of
mathematical knowledge. Ernest (1991) mentioned that the
absolutist nature of mathematics portrays knowledge as certain
and unchangeable truth, mathematical knowledge is made up of
absolute truth and represent the unique realm of certain
knowledge
• Moreover, absolutist nature of mathematical knowledge is based
on axioms and definition (Ernest, 1991) which is fixed,
unchangeable and objective
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Contd..
• My beliefs towards mathematics were changed during my masters
in mathematics education
• I came to know various pedagogies (activity based instruction,
collaborative and cooperative approach, critical pedagogy, etc.)
that has been effective in my mathematics class
• Those kinds of pedagogies promote more engaging, practical as
well as meaningful learning
• More Fallibilist in nature
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Contd…
• Ernest (1991) mentioned that mathematical truth is fallible and
corrigible and open to revision. It also challenges the absolutist
nature of mathematics. Fallibilist nature of mathematics focused
on revision, revisiting, and claimed that knowledge is subjective
• Mathematics as human inventions rather than discovery
• The major source of fallibilist nature of mathematics is
Wittgenstein’s philosophy where he described mathematics as a
language game (Lerman, 1990)
• Mathematics is associated to our day to day communication and
interaction
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Contd…
• Similarly, Luitel (2012) discussed the nature of mathematics as
im/pure knowledge
• It helps to empower the inclusive nature of mathematical
knowledge
• Impure knowledge in mathematics focuses on informal
mathematics, cultural artifacts, local mathematical knowledge,
etc. which is falliblist in nature and pure knowledge in
mathematics is absolutist in nature
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Contd…
• D’Ambrosio (2001) mentioned that
ethnomathematics is use to express
the relation between culture and
mathematics
• His view has been focused on
cultural nature of mathematics
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Source: Google Image

Contd…
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Now a days it is popular in Nepal. More than one
hundred different ethnic groups and many of them
have rich mathematical practices in their day to day
practices (Pant & Luitel, 2016)

Contd…
• Loveridge et.
al (2005)
articulated
the different
views of
nature of
mathematics
given by the
students.
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Math is not just about numbers; math is something
that you can make really fun, especially with
geometry and symmetry, because you can draw shapes
and draw characters that you
like. (Year 5–6)
Patterns because plus is minus and plus is times and
times is division and division are fractions and
fractions are decimals and decimals are percentages
and it goes on and on (Year 2– 4).
Math is like something you use every day (year 7 – 8).

Historical Background of the Nature of
Mathematics
• It has been discussed even
before the fourth century, Plato
and Aristotle are the first who
provide the space to discuss
the nature of mathematics.
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Contd…
• From Plato’s point of view, objects of mathematics had an existence of
their own, behind the mind, in the external world
• From Aristotle’s point of view mathematics were not based on a theory
of an external, independent, unobservable body of knowledge but were
based on experienced reality where knowledge is obtained from the
experimentation, observation and abstraction
(Dossey, 1992)
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Contd…
• To deal with the problems related to nature of mathematics new views
of mathematics arose, namely the school of logicism, school of
institution, and the school of formalism (Dossey, 1992)
• School of logicism is more related to the Plato’s point of view of
mathematics
• Rotman (2006) mentioned that Platonist views of mathematics is
neither a formal and meaningless games nor some kind of languages
and mental construction, but a science, a public discipline concerned
to discover and validate objective or logical truth
• Logicism claims that all the mathematical truth can be proved from the
axioms and the rules of inference of logic alone (Ernest, 1991).
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Contd…
• Intuitionist accepts only the mathematics that could be developed
from the natural numbers forwarded through the mental activities of
constructive proofs.
• Intuitionist picture of mathematical assertation and proofs depends
on the coherence and acceptability of what it means by an effective
procedure (Rotman, 2006).
• A certain mental creation has been carried out
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Contd…
• For the formalist, mathematics is a
species of game, a determinate play
of written marks that are transformed
according to explicit unambiguous
formal rules (Rotman, 2006)
• Formalist claim that pure
mathematics can be expressed as
uninterpreted formal systems, in
which truths of mathematics are
represented by formal theorem
(Ernest, 1991)
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Pedagogical Practices in Mathematics Classroom
• Thompson (1992) mentioned that, person’s understanding of the
nature of mathematics predicates that person’s view of how teaching
should take place in the classroom.
• Pedagogical practices of teachers’ depend on their beliefs and
perception towards mathematics.
• Presmeg (2002) has argued that beliefs about the nature of
mathematics either enable or constrain the bridging process between
everyday practices and school mathematics (as cited in Loveridge et
al., 2005)
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Conclusion
• It is difficult to define what mathematics is, it can be seen and
observed from the different angle
• It also depends on the persons’ experiences in the field of mathematics
and mathematics education
• However, Fallibilist , impure and culture nature of mathematics
enables to provide inclusive education and provides equity and justice
for marginalized groups of students as well as enables teacher to
incorporate humanizing pedagogy
• The images of mathematics curriculum is guided by; curriculum as
experience, curriculum as agenda for social reconstruction, curriculum
as currere (Schubert, 1986)
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References
• Dossey, J. (1992). The nature of mathematics: Its role and its influence. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and
learning (pp. 39–48). New York: Macmillan.
• Ernest, P. (1991). The philosophy of mathematics education. Taylor and Francis Group: London.
• Jha, k., Adhikary, P.R., & Pant, S.R. (2006). A history of mathematical sciences in Nepal.Department of Natural Science and Mathematics, School of
Science, Kathmandu University, P.O. Box No. 6250, Dulikhel, Nepal.
• Lerman, S. (1990). Alternative perspectives of the nature of mathematics and their influence on the teaching of mathematics. British educational
research journal. 16(1), 53-61. Taylor & Francis.Loveridge, Y. J., Taylor, M., Sharma, S. & Hawera, N. (2005). Students’ perspectives on the nature of
mathematics.
• Luitel, B. C. (2012). Mathematics as an im/pure knowledge system: Symbiosis (w)holism and synergy in mathematics education. International Journal
of Science and Mathematics Education 10(6). Taiwan: Springer
• Pant, B. & Luitel, B. C. (2016). Beliefs about the nature of mathematics and its pedagogical influences. Presented on 13th international conference on
mathematical education, Humburg, 24-31, July, 2016.
• Rotman, B. (2006). Toward a semiotics of mathematics. In Hersh, R. (Ed) 18 Unconventional Essays on The Nature of Mathematics. Springer.
• Schubert, H. W. (1986). Curriculum: Perspective, paradigm and possibility. Newyork: Macmillan.
• Stinson, D. W., Bidwell, C. R., & Powell, G. C. (2012). Critical pedagogy and teaching mathematics for social justice. International journal of critical
pedagogy, 4(1), 76-94.
• Thompson, A.G. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. In D.A. Grouws(Ed.), Handbook of research on mathematics
learning and teaching, (pp.127–146). New York: Macmillan Publishing.
• Thurston, W. P. (2006). On proof and progress. In Hersh, R. (Ed.) 18 Unconventional Essays on The Nature of Mathematics. Springer.
3/24/2019
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Thank You !!!
***Q/A***
3/24/2019
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