Power point on chapter 2 for B.Ed methodology of teaching mathematics course

1. 1Angel Rathnabai

2. Mathematics????
2Angel Rathnabai

3. Some Math Myths
Math = terrifying
Math = static (Greeks,
Newton,…)
Math other sciences
Math = solitary
Math = impractical major as
career preparation 3Angel Rathnabai

4. Math has an Image Problem
Mathematicians — smart, but “from
another world”
Actually, math lurks behind the
curtain of popular subjects:
simulation, forecasting, data mining,
networks, polling, design,
optimization, synthetic environments
Mathematicians have no monopoly on
the practice of math — nearly
4Angel Rathnabai

5. Intellectual Foci of the Sciences
Field The Study of …
Astronomy Origin and evolution of stars,
galaxies, and the universe
Biology Genomics, organisms, ecologies
Chemistry Molecular structure of matter
Earth Sciences Geology, Oceanography,
Atmospheric Sciences, Geophysics
Materials Science Micro- and macrostructure of
materials
Physics Space-time, energy, atomic and
nuclear structure of matter
Mathematical Sciences Patterns, structures, abstract models
of reality
5Angel Rathnabai

6. Views about
Mathematics
6Angel Rathnabai

7. Student’s View of mathematics
(Schoenfeld, 1992)
Mathematics problems have one and only one right answer.
Mathematics is facts and rules with one way to get the right
answer. You find the rule and get the answer. Usually, the rule
to use is the one your teacher just taught you.
You don‟t need to understand why the rules work.
If you don‟t solve a problem in five minutes, then you‟ll never
solve it. Give up.
Only geniuses discover or create mathematics, so if you forget
something, you‟ll never be able to figure it out on your own.
Mathematics problems have little to do with the real world. In
the real world, do what make sense. In mathematics, follow the
rules.
Mathematics is arithmetic
7Angel Rathnabai

8. Parent’s View of mathematics
(Schoenfeld, 1992)
Mathematics is about numbers and arithmetic,
unbending accuracy and infallible rules.
The students should know the basics.
Mathematics is an innate ability. Mathematics is
difficult, and so, students should not be expected to
do too much.
8Angel Rathnabai

9. Teacher’s View of mathematics
Richard Skemp (1976) : there are two effectively
different subjects being taught under the same name
“mathematics”.
Instrumental Mathematics
It consists of a limited number of rules without
reasons
Relational Mathematics
It is knowing both what to do and why. It involves
building up conceptual structures or schemas
from which a learner can produce an unlimited
number of rules to fit an unlimited sets of
situation.
9Angel Rathnabai

10. Comprehensive View of Mathematics
Mathematics is not arithmetic.
Mathematics is problem posing and problem
solving.
Meaningful problems take a long time to pose as well as to solve.
They stimulate curiosity about mathematics, not just about the
answer to a problem. They engage a variety of students‟ ideas and
skills. They lead students to thinking about how the world work from
a mathematical point of view and to think about how mathematics
itself works. They open up discussion to a variety of contributions
from multiple participants.
Cont…
10Angel Rathnabai

11. Mathematics is the activity of finding and studying
patterns and relationships.
Mathematical activity includes perceiving, describing,
discriminating, classifying, and explaining patterns
everywhere in number, data, and space, and even in
patterns themselves.
Mathematics is a language.
Mathematics is also used to communicate about
patterns.
Mathematics is doing mathematics.
The process of „doing‟ mathematics is far more than just
calculation or deduction; it involves observation of patterns,
testing of conjectures, and estimation of results.
Cont…
11Angel Rathnabai

12. Mathematics is a path to independent thinking.
Mathematics is an area in which even young children
can pose and solve a problem and have confidence
that the solution is correct not because the teacher
says it is, but because its inner logic is so clear.
Mathematics is a way of thinking and a tool for
thinking.
Mathematics is a changing body of knowledge, an
ever-expanding collection of related ideas.
Mathematics is useful for everyone.
Cont…
12Angel Rathnabai

13. Meaning of
Mathematics
13Angel Rathnabai

14. Mathematics – Meaning
„máthēma‟ - "that which is learnt", "what one gets to
know"
máthēma is derived from „manthano‟, while the modern
Greek equivalent is „mathaino‟ - "to learn“
adjective is „mathēmatikós‟ meaning "related to
learning" or "studious“
In Latin, and in English until around 1700, the term
mathematics more commonly meant "astrology" (or
sometimes "astronomy") rather than "mathematics"; the
meaning gradually changed to its present one from
about 1500 to 1800.
In English, the noun mathematics takes singular verb
forms. It is often shortened to maths or, in English-
speaking North America, math. 14Angel Rathnabai

15. Definition of
Mathematics
15Angel Rathnabai

16. Definition – Mathematics
„“The abstract science which investigates deductively the
conclusions implicit in the elementary conceptions of spatial
and numerical relations, and which includes as its main
divisions geometry, arithmetic, and algebra”
- Oxford English
Dictionary, 1933
“The study of the measurement, properties, and relationships
of quantities and sets, using numbers and symbols”
- American Heritage
Dictionary, 2000
“The science of structure, order, and relation that has evolved
from elemental practices of counting, measuring, and
describing the shapes of objects”
- Encyclopedia
Britannica
16Angel Rathnabai

17. Definition – Mathematics
“The science of quantity”
- Aristotle
“The science of indirect measurement” (1851).
- Auguste Comte's
Three leading types of definition of mathematics are called:
O Logicist,
“The science that draws necessary conclusions" (1870).
- Benjamin Peirce's
O Intuitionist,
"All Mathematics is Symbolic Logic"(1903).
- Russell's
O Formalist,
"Mathematics is the mental activity which consists in carrying out
constructs one after the other“
- L.E.J. Brouwer
17Angel Rathnabai

18. Definition – Mathematics
“The subject in which we never know what we are talking about, nor
wheather what we are saying is true”.
- Bertrand Russell (1901)
“Mathematics is the indispensible instrument of all physical researches”.
- Kant
“Mathematics is the queen of sciences and arithmetic is queen of all
mathematics”.
- Gauss
“Mathematics is the gateway and key to all sciences”.
- Bacon
“Mathematics is the language of physical sciences and certainly no more
marvelous language was created by the mind of man”
- Lindsay
“Mathematics is a way to settle in the mind a habit of reasoning”.
- Locke
“Mathematics is engaged, in fact, in the profound study of art and the
expression of beauty”.
- J.B.Shaw
18Angel Rathnabai

19. Nature of
Mathematics
19Angel Rathnabai

20. Nature of
Mathematics
A science
of
discovery
An
intellectual
game
The art of
drawing
conclusion
s
A tool
subject
A system
of logical
processes
An intuitive
method
20Angel Rathnabai

21. Characteristics of
Mathematics
21Angel Rathnabai

22. Characteristic
s of
Mathematics
Logical
sequence
Structure
Precision and
accuracy
Abstractness
Mathematical
Language and
Symbolism
Applicability
Generalisation
and
classification
22Angel Rathnabai

23. Development of
Mathematics
23Angel Rathnabai

24. An Ancient Subject
Pythagorus of Samos
569-475 B.C. (?)
musician, geometer,
first “pure” mathematician
24Angel Rathnabai

25. A Modern Subject
Karen Uhlenbeck, 1942 - University of
Texas
Partial differential equations and
mathematical physics
National
Academy of
Sciences
National
Medal of
Science, 2000
25Angel Rathnabai

26. Major Subfields of Math Sciences
Subfield The Study of …
Foundations Axiomatic underpinnings of mathematics
Algebra and Combinatorics Structures, relations between discrete objects
Topology and Geometry Spatial structures, patterns, shapes
Number Theory Properties of numbers and polynomials
Analysis Theory of functions, extensions and generalizations of the
calculus
Probability Randomness and stochastic phenomena
Statistics Collection, analysis, and application of data
Applied Mathematics Modeling, analyzing and optimizing systems
Computational Mathematics Computer-based, experimental mathematics
26Angel Rathnabai

27. Math as the Hub of Science
Mathematical
Sciences
Physical
Sciences
Chemistry Biological
Sciences
Engineering
and
Technology
Agricultur
e
Economics
Psycholog
y
Finance
Communicatio
n
Geoscience
s
Computers
Arts
MusicArcheolog
y
Logic
Philosoph
y
Linguistic
s
Managemen
t
Geograph
y
27Angel Rathnabai

28. Math in Society and Technology
Problem/Application Contribution from Mathematics
MRI and CAT Imaging Integral transforms, geometry
Internet: search engines, compression Graph theory, linear algebra, wavelets
Financial options valuation Black-Scholes model and Monte Carlo simulation
Global reconnaissance Signal processing, image processing, data mining
Confidentiality and integrity Number theory, cryptology/combinatorics
Modeling of atmosphere and oceans Wavelets, statistics, numerical analysis
Analysis of the human genome Data mining, pattern recognition, discrete
algorithms
Rational drug design Data mining, statistics, optimization
Digital entertainment, animation Signal processing, geometric and graphic algorithms
Aerodynamic design Differential equations, optimization
Earthquake analysis and prediction Statistics, dynamical systems/turbulence
28Angel Rathnabai

29. Few Applications
of Mathematics
29Angel Rathnabai

30. Applied Math - What’s playing in
classrooms and labs near you?
Functional Genomics
Finance
Cryptography
Simulation of physical systems (e.g.,
airplanes, tokomaks, hurricanes)
Simulation of discrete systems (e.g., traffic
flow, networks, battlefields)
Immersive visualization
30Angel Rathnabai

31. Functional Genomics
The Protein Folding Problem
sandwich
protein
immunoglobulin
PDB: 7FAB
31Angel Rathnabai

32. Image Processing: Inpaintings
Scratch
Removal
Disocclusion
Graffiti Removal
32Angel Rathnabai

33. Synthetic Images
33Angel Rathnabai

34. What’s Math got to do with it
Picture courtesy
of Doug Roble,
Digital Domains.
34Angel Rathnabai

35. Simulation Example: Aerodynamics
Airflows over wing models can be computed and visualized
1999 Gordon Bell Prize
35Angel Rathnabai

36. Simulation Example: Bioinformatics
Proteins fold in a way that minimizes configuration energy
36Angel Rathnabai

37. “Experimental Mathematics”
"There will be opened a gateway
and a road to a large and
excellent science into which
minds more piercing than mine
shall penetrate to recesses still
deeper."
Galileo (1564-1642) on “experimental mathematics”
37Angel Rathnabai