Appreciationof mathematics:My observations and opinions

Appreciation of
Mathematics: Current
Scenario in my
Opinion

Welcome to All Participants
Prof NB Venkateswarlu
Professor, AITAM, Tekkali
Visakhapatnam
venkat_ritch@yahoo.com
www.ritchcenter.com/nbv

Let Me first
Congratulate all
the Organizers.

What am I going to talk?
• Status of Mathematical teaching in
Engineering- In my opinion
• Retrospection of possible reasons for the
prevailing pathetic situation.
• My perception about teaching
mathematics to Engineering Students
• My views about how to correct in a
humble manner

Remember I am only going to
share my experiences and
observations.
I am neither a Mathematician nor
a Computer Scientist. I try to be
an Engineer first though I know I
am still half-baked Engineer.

I got enlightenment about
Computer Science after reading
the book
Computational Geometry,
Preparata, Springer Series.

Also, I want to remind you I am
not going to be a fool by
promising that I can talk about
whole mathematics useful for
Engineers.
Only an Iota of it I shall
expose.

I love my High School Mathematics
teacher even now.
“Students’ understanding of mathematics,
their ability to use it to solve problems, and
their confidence in, disposition toward,
mathematics are all shaped by the
teaching they encounter in school”
(NCTM, 2000, p.16-17).

MMaatthheemmaattiiccss iiss
FFaasscciinnaattiinngg!!!!
Many discoveries in Physics
were predicted by Mathematics
before they were observed
experimentally.

Radio Waves, Big Bang Theory,
General Theory of Relativity, Planck’s
quanta, Black Holes, Antimatter,
Quarks,…
Ref: Mathematics as a Sixth Sense, Stephane Durand
http://www.math.ecnu.edu.cn/earcome3/poster/EARCOME3_Durand_Stephane_Poster.d
oc
EExxaammpplleess

According to ACM 2001Committee
A computer Science students should
posses a certain level of mathematical
sophistication such as:
• Ability to formalize concepts
• Work from definition
• Think rigorously
• Reason correctly
• And construct a theory

What does it take to become an
engineer?
• Mathematics
• Science
• Creativity

What is Engineering?
• What do engineers do?
• Engineers design and build things.
• Engineers create technology.
• Engineering is different from Science.

• Science is the study of what is.
• Engineering is the creation of
what is to be.

Engineering is different from
science.
• Science
– Discovery
– Understanding
– Knowledge
– Natural world
– “The world as we
found it”
• Engineering
– Design
– Creating/producing
– Technology
– Artificial world
– The world we create

Design
• The man-made world
• The creation of artifacts
• Adapting the environment to our needs
and desires
• Concern of engineers, architects, and
artists

Design as problem solving
• Given
– Problem specification
– Initial conditions
– Constraints
– Standards/regulations
• Find a Solution

Design is creative
• Design problems
– Open-ended
– Ill-defined (vague)
– Multiple alternatives
– Generate lots of solutions

Design is Experimental and
Iterative
• Getting it right takes many tries
• The first cut is rarely good enough
• Some designs fail
• Even if satisfactory, most designs can be
improved
• Once it works, refine it

Design cycle
• Requirements, problem
• Generate ideas
• Initial concept
• Rough design
• Prototype
• Detailed design
• Redesign

Design
• The core problem solving process of
technological development
• “It is as fundamental to technology as
inquiry is to science or reading is to
language arts”

Serious Problems in Science, Technology,
Engineering and Math Education
• Declining enrollments in engineering
programs
• Numbers of women and minority students in
engineering are not representative of general
population
• Lower science and math test scores of high
school students with respect to the rest of the
industrial world
• Technological illiteracy

Whom we have to blame for this
worst situation?
• Parents
• Students
• Industry
• Universities or other controlling authorities
• College managements
• Lastly, faculty

I have illustrated problems related
to parents, managements,
Universities in my lecture hosted at:
http://www.slideshare.net/venkatritch/peda
gogy-in-engineering-colleges

Blockages
Why don’t more people do Mathematics?

BBlloocckkaaggeess
Mathematics is hard!
Yes it is! But it is also very rewarding,
and is no more harder than learning to
skate or tennis! It takes time to
understand new ideas and concepts.
In any endeavour you need to do
something hard to excel!

BBlloocckkaaggeess
You need to be bright to do
Mathematics.
No! You need not be very bright. But
Mathematics makes your brighter. And
it will improve your skills and
understanding of other related
subjects.

BBlloocckkaaggeess
I don’t need a lot of Mathematics for
science!
Wrong! A higher level of Mathematical
skill will make you a better Scientist
and Engineer.
Great discoveries and higher level
performance in physics and
engineering innovation requires high
level Mathematics.

Rewards of doing Mathematics
• Problem solving skills that will help you in
every aspect of your life.
• Good organisational skills.
• Logical, clearer thinking.
• A very interesting, satisfying life full of
challenges and achievements!

By the way who are our
Students?

Who are our PhD students and
faculty vice versa?
Computers are great tools, however, without
fundamental understanding of engineering problems,
they will be useless.

Who are Our Students?:My
observations
• They put face that they did not hear
compound interest at all.
• If you probe further and throw hits and use
patting words, now some of their faces
glows.
• If you insist further, the answer is “sorry
we don’t remember the equation”.
• Some may write for final amount, but not
to interest.
P*(1+r/100)^t-P

Who are Our Students-Cont
• Simple interest =P*T*R/100
• If I say R is in ratio instead of percentage,
then also they don’t understand how to
change the equation.
• Of course, majority of them have 150 out
of 150 in Mathematics in their 10+2.

Do ask them about our
Intermediate Example on
Simple Pendulum
• Why do we draw the line?.
• To forecast g value at our place

An example: Grade to points
(They don’t have analysis skills.
They wait for answer for a
problem)
Grade Points
A (65) 10
B (66) 8
C (67) 6
D (68) 4
E (69) 2

They find very difficult to relate
to mathematics.
Answer: P=2*(70-g)
(65,10)
(69,2)
g
P

One More example from an US
based high school competition.
• Given a capital letter we need to find another upper case letter that
is d units from the given letter. You need to count cyclically.
• The following table is for a d value of 4
Input capital letter
with its ASCII code
Output capital letter
with its ASCII code
A(65) E(69)
E(69) I(73)
F(70) J(74)
V(86) Z(90)
W(87) A(65)
X(88) B(66)
Y(89) C(67)
X(90) D(68)

Not even 1% can think of
converting degrees in radians to
degrees, minutes and seconds.
• They don’t even remember how many
seconds makes a minute
• They don’t perceive that angle can be
more than 360 degrees.

Did you ever ask how they can
convert a given temperature in
one scale to all other scales.
• At most 40% can recall.
• Only 10% recollects 273.03 correctly.

They take more time to related
to The World examples.
• Speed, Distance and Time
• Small examples involved bits, bytes, bps,
etc., is too confusing for them.
• May be a mathematics teacher has to
change from distance, time, speed to bits,
time and Mbps in the beginning itself

They find hard to relate to
mathematics.
How many digits are there in a
given integer?
What is the largest integer
which is integer power to 10 and
divides a given integer?

Guess from the following data?
Recall the definition of
logarithm.

log10(10)=1
log10(99)=1.99999999
log10(100)=2
log10(999)=2.99999999
log10(1000)=3
log10(9999)=3.99999999
log10(10000)=4
log10(99999)=4.99999999
log10(100000)=5
log10(999999)=5.99999999

They feel hard to understand
number of bits versus
logarithms?.

A practical example to illustrate
use of logarithms, simultaneous
equations. We want them to
appreciate mathematics and
develop interest in it. May be, I
am of the opinion is that to give
live examples as many as
possible to elucidate a concept.

• Dual-energy X-ray absorptiometry (DEXA).
This is the most accurate way to measure
BMD. It uses two different X-ray beams to
estimate bone density in your spine and hip.
Strong, dense bones allow less of the X-ray
beam to pass through them. The amounts of
each X-ray beam that are blocked by bone and
soft tissue are compared to each other. DEXA
can measure as little as 2% of bone loss per
year. It is fast and uses very low doses of
radiation but is more expensive than
ultrasound testing.
• http://www.webmd.com/osteoporosis/bone-mineral-density

• Calculation of Bone Mineral Density:
• The basic equations for dual-photon
absorptiometry can be derived from a number of
underlying assumptions. First, it is assumed
that the material is composed of varying
amounts of only two substances (in this case
bone and soft tissue). Second, it is assumed
that scatter can be ignored. Under these
circumstances, for any given photon energy, the
number of photons striking the detector (N) can
be calculated from the number of incident
photons (No) using Beer’s Law.

• Beer’s Law:
exp ( ), o s s b b N = N - m M +m M
where μs and μb represent the mass attenuation
coefficients (cm2/g) of soft tissue and bone
(respectively) and Ms and Mb represent the area
densities (g/cm2) of the two tissue types. If data are
acquired at two different energies and the above
equation rearranged, a set of two equations with two
unknowns is generated as follows:

ln(N ) = m +m
OL M M
N
sL s bL b
L
ln(N ) = m +m
OH M M
N
sH s bH b
H
where the subscripts L and H have been added to
distinguish the low- and high-energy data sets.
The two unknowns are Ms and Mb and the above
pair of equations can be solved for either quantity
using the method of simultaneous equations
(systems).

How to correct the situation?
• There can be hundreds of ways to correct.
Out of all, teaching mathematics should
be carried out with real life examples.
Preferably introduce feel of Engineering
along with the example. Of course, for this
to happen, mathematical faculty has to
enrich themselves with engineering
applications. Of course an Engg. Faculty
has to work in other way wrong. I
understand some UK university has
started a course “Mathematical
Engineering”.

My views on some mathematical
concepts and possible live
examples to be introduced.
• Geometry
• Calculus
• Algebra
• Trigonometry

Fitting Line – Least Squares
Approach

Linear Classifiers
x f yest
denotes +1
denotes -1
f(x,w,b) = sign(w x + b)
How would you
classify unknown
data?
w x + b=0
w x + b<0
w x + b>0

Computer Graphics – Drawing a
Line

Area under a curve.
• Where is it practically used?
• In Civil Engg to calculate volume of cutting
and filling.

Air Pillows In Car to save
humans
• Head Injury Index (HIC) – Crash test and
air bags

Severity Index
• The first model developed historically was the Severity Index (SI).
• It was calculated using the formula:
• The index 2.5 was chosen for the head and other indices were used
for other parts of the body (usually based on possibly gruesome
experiments on human or animal bodies).
• The Severity Index was found to be inadequate, so researchers
developed the Head Injury Criterion ».

Head – Simple Pendulum
Motion

Braking
• Normal braking in a street car: 10 ms-2 (or about 1 g).
• Normal braking in a racing car: 50 ms-2 (or about 5 g).
This is due to aerodynamic styling and large tyres with
special rubber.
• When we stop in a car, the deceleration can be either
abrupt (as in a crash), as follows:
• or more gentle, as in normal braking:
• Either way, the area under the curve is the same, since
the velocity we must lose is the same.

Crash Tests
• Imagine a car travelling at 48.3 km/h (30
mph). Under normal braking, it will take
1.5 to 2 seconds for the car to come to
rest.
• But in a crash, the car stops in about 150
ms and the life threatening deceleration
peak lasts about 10 ms.

A3-ms value
• The A-3 ms value in the following graphs
refers to the maximum deceleration that
lasts for 3 ms. (Any shorter duration has
little effect on the brain.)

• If an airbag is present, it will expand and
reduce the deceleration forces. Notice that
the peak forces (in g) are much lower for
the airbag case.

• The blue rectangles in these deceleration
graphs indicate the most critical part of the
deceleration, when the maximum force is
exerted for a long duration.
• With an airbag, you are far more likely to
survive the crash. The airbag deploys in
25 ms.

Golden Ratio: Phi
Parthenon Greece

Leonardo da Vinci's "Vitruvian Man",
showing the golden ratio in body dimensions

Golden Ratio: Beauty’s Secret

Silver Ratio
Pell numbers: 1, 2, 5, 12,29
Silver ratio=1+sqrt(2)

Triangulation
• c= light speed
• ts=receiver clock offset time

An Image Processing
Example: IP and CG are
complimentary

Original, directional, Laplacian,
Sharpening

An excellent example to
illustrate the use of orthogonal
vectors.
CDMA: Code Division Multiple
Access which is used in cell
phones, satellite phones, and
vice versa.

CDMA
• One channel carries all transmissions at
the same time
• Each channel is separated by code

CDMA: Chip Sequences
• Each station is assigned a unique chip sequence
• Chip sequences are orthogonal vectors
– Inner product of any pair must be zero
• With N stations, sequences must have the
following properties:
– They are of length N
– Their self inner product is always N

An excellent example to
illustrate the use of orthogonal
vectors.
CDMA: Bit Representation

Sequence Generation
• Common method: Walsh Table
– Number of sequences is always a power of two

How to teach rotation,
translation, etc with live
examples?

Operations of Photographs?
• Scaling
• Zooming
• Rotation
• Translation
All the above can be nicely introduced by
taking a simple image and using MATLAB
or paint or GIMP. Why a mathematics
teachers tries to be too abstract?

Example use in Robotics:
Kinematics and Dynamics.
Kinematics: Direct Kimematics: If we
apply a series of rotations and
translations where will be the robot
gripper? Inverse Kinematics: Also, what
rotations have to be applied at each
joint to position at a position. Dynamics
deals with stability of Robot.

Astronomy involves full of
rotations and transformations.

Estimating 3D information Two
Snaps – Binocular Vision.
It does involves number of
transformations.

Standard Deviation?. What for?
• Example of Production Process (Quality
Control Engineers)
• ఫైైవ్ సాట్ర్ హొటల్ కు వైళేళ్ది
ఎ0గిలి కూడు తినడానికా?. నిజమే. There
will be a taster, we takes a piece of the
prepared item and only if it tastes good he
will be sending for serving.
• Analyzing students marks of an
examination Center
• A companies share

What is the practical use of
Correlation?
• Hardly very few really relates.

Finite differences: relation
estimation from the observed
data on independent and
dependent variables.

Newton Raphson Method
• Sqrt() function of C language

What is a Determinant?.
An example from statistics. In
multivariate statistics,
covariance matrix represent
spread of points in the multi-dimensional
space. If
determinant is small then
samples are compact, otherwise
spread widely.

What are actually Eigen Values
and eigen vectors?.

Childhood Game
A man with Tiger, Goat, and gross packet
wanted to cross a river. The boat can
carry two people at a time. What are the
steps he has to follow?.

Queens Problem
Do They Hit each other?.

Recall “Stallin” Cinema
• If a fellow helps 3 people, and those three helps
3 each, and further they help three more, how
many
1+3 + 3*3 + 3*3*3 + 3*3*3*3 + …… 3^r =
= ½ * 3^(r+1) -1
If r=16 the sum is 6,45,70,031

Deadlocks in Networks
• Same as accidents on Roads

Search Engineer – To Divert the
Internet Traffic to Our Site

Click Based Charging –
AdWords of Google and Yahoo

Tsunami Warning Systems.
• How many models?. About 120 models.

A physics problem illustrated
mathematically. Why we can not do
in the same way in our class?

D U F = F + F
Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
117
Newton’s 2nd law of Motion
• “The time rate change of momentum of a body is
equal to the resulting force acting on it.”
• Formulated as F = m.a
F = net force acting on the body
m = mass of the object (kg)
a = its acceleration (m/s2)
• Some complex models may require more sophisticated
mathematical techniques than simple algebra
– Example, modeling of a falling parachutist:
FU = Force due to air resistance = -cv (c = drag
coefficient)
FD = Force due to gravity = mg

Independent variable
Dependent variable
v(t) = gm 1- -( / )
F
dv
=
F = F +
F
F mg
D
=
F = -
cv
mg cv
m
dv
dt
m
dt
U
D U
= -
dv = -
g c
v
m
dt
• This is a first order ordinary differential equation.
We would like to solve for v (velocity).
• It can not be solved using algebraic manipulation
• Analytical Solution:
If the parachutist is initially at rest (v=0 at t=0),
using calculus dv/dt can be solved to give the result:
( e c m t )
c
Forcing function Parameters

v(t) = gm 1- -( / )
119
Analytical Solution
( e c m t )
c
t (sec.) V (m/s)
0 0
2 16.40
4 27.77
8 41.10
10 44.87
12 47.49
∞ 53.39
If v(t) could not be solved analytically, then
we need to use a numerical method to solve it
g = 9.8 m/s2 c =12.5 kg/s
m = 68.1 kg

Numerical Solution
dv
v t v t
= -
( ) ( ) ........ lim
v
= D
This equation can be rearranged to yield
v t v t g c i i = + - - + +
120
i i
g c
v
dv
@ D
v t -
v t
( ) ( ) i i
( )
1
1
0
1
1
i
i i
t
i i
v t
m
t t
t
dt
t t
t
dt
= -
-
D
-
D
+
+
D ®
+
+
( ) ( ) [ ( )]( ) 1 i i 1 i v t t t
m
Δt = 2 sec
To minimize the error, use a smaller step size, Δt
No problem, if you use a computer!
t (sec.) V (m/s)
0 0
2 19.60
4 32.00
8 44.82
10 47.97
12 49.96
∞ 53.39

vs. Numerical solution
t (sec.) V (m/s)
0 0
2 19.60
4 32.00
8 44.82
10 47.97
12 49.96
∞ 53.39
Δt = 0.5 sec
Δt = 0.01 sec
v t v t g c i i i ( + 1) = ( ) +[ - ( )]D
v(t) = gm 1- -( / ) v t t
m=68.1 kg c=12.5 kg/s
g=9.8 m/s
t (sec.) V (m/s)
0 0
2 16.40
4 27.77
8 41.10
10 44.87
12 47.49
∞ 53.39
( e c m t )
c
m
Δt = 2 sec
Analytical
t (sec.) V (m/s)
0 0
2 17.06
4 28.67
8 41.95
10 45.60
12 48.09
∞ 53.39
t (sec.) V (m/s)
0 0
2 16.41
4 27.83
8 41.13
10 44.90
12 47.51
∞ 53.39
CONCLUSION: If you want to minimize
the error, use a smaller step size, Δt

My views about how to correct
in a humble manner
• Let Professors of IIT’s or IISC’s or ISI or
Chennai Institute of Mathematics or TIFR
to form faculty interest groups and groom
them with necessary inputs to teach
mathematics more effectively in colleges. I
remember an example situation related to
Nanotechnology. I read some where that
what first Taiwan Government did is to
develop 5 to 10 examples to be taught at
school level to introduce Nanotechnology.
They did not grant research funds first!.

My views about how to correct in a
humble manner
• Build awareness among Mathematics
people about Engineering examples.
• Encourage combined lesson
development with excellent Engineering
examples by both mathematics and
engineering faculty.
• Develop teaching tools/models/prototypes.
• Encourage students to appear for
Mathematics Olympiad, Informatics
Olympiad.
• Organize local/regional competitions.

Some of my efforts towards
encouraging competitions.
• I am maintaining ICPC examples in my personnel web
site since 15 years.
• I tried to motivate mathematics faculty. A small but
positive response from GVP Womens’ college.
• Wrote a book which under print titled “101 Programming
Problems solved: Join us to win Informatics Olympiad”.
• I tried to convince Sri Vishnu Raju garu also.
• I am trying to motivate high school teachers around
Visakhapatnam.
• I requested former AU Registrar Prof Prasada Reddy
garu to recommend to some suitable colleges.
• I approached CSI people.

ToVishnu Raju BVRCEW Oct 22, 2013
Dear Sri. Raju Garu,
How are you?. Hope you remember me.
I am writing this letter to explore the possibility of initiating student orientation programs to an
international competition at your engineering colleges which are at Bhimavaram and Hyderabad.
Since 1970’s, Association of Computing Machinery (ACM) an international voluntary body and IBM
have initiated an international programming competition under the name hood of “International
Collegiate Programming Contest(ICPC)”. Only from last ten years, few Indian institutes such as IIT-K,
IIT-Kg, IIIT-Hyd, Amrutha Univ, are participating.
I feel it is high time for institutes such as yours who are thriving for excellence to take steps to
orient your students to participate in ICPC. Also, students can participate in other world level
contests such as challenge24, Microsoft Cup, etc. In addition, they can take part in some Indian
contests organized by Infosys, Wipro, etc.
Being an active teacher in computer science for more than 25 years, I would like to groom your
students for the above examinations. In this connection, I would like to discuss with you. As I know
that you often visit’s Visakhapatnam, I request you to give appointment to me in your next visit to
Visakhapatnam so that I can explain my ideas in detail in person.
I am looking forward for your response.
With best regards
Prof NB Venkateswarlu

My views on correcting the situation
• Is it possible to reduce class strength to
20-25?
• Is it possible to send faculty to class only
after orienting them to dogma of teaching?
• Is it possible to send only qualified faculty
to a course. In 4th year level, “electives” are
taught by just passed faculty. Where as in
IIT’s, unless a senior professor of that
specialization retires, the next senior will
not get chance to teach that elective. What
a fun taking place in our colleges?

My Views - Continued
• Project Expos by Mathematics and
Engineering departments.
• Seeing Engineering question papers to
have at least 30-40% of questions
involving mathematics.
• Maintaining a repository of live examples
and maintaining the same like the
following.

Useful websites
• http://integralmaths.org
• http://www.teachengineering.org
• http://www.tryengineering.org
• http://www.intmath.com
• http://pumas.jpl.nasa.gov
• http://pumas.gsfc.nasa.gov
• http://www.citrl.net
• http://www.mathsisfun.com

See an exemplary explanatory
lesson prepared by
www.integralmaths.org • http://integralmaths.org/pluginfile.php/842
94/mod_resource/content/0/AirTrackingTe
acherFinal.pdf
AirTrackingPresentation

My Views-Cont
I remember my 10+2 teacher Mr John
Wilson mentioning “What he can teach to
us what he has learned in his Masters”?
Some how I am of the opinion that last 25-
30 years in India, 10+2 syllabus is not
revamped. I am the first batch student of
1000 marks. Since then no major changes
has taken place. Otherwise tremendous
developments taken place in
mathematics. Unless we do something,
the developments can not be passed
down to generations.

Appreciationof mathematics:My observations and opinions

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