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

[email protected]

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.

“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).

I love my High School Mathematics teacher even now.

Many discoveries in Physics

were predicted by Mathematics before they were observed experimentally.

Mathematics is Mathematics is Fascinating!!Fascinating!!

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.doc

ExamplesExamples

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/pedagogy-in-engineering-colleges

Blockages

Why don’t more people do Mathematics?

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!

BlockagesBlockages

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.

BlockagesBlockages

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.

BlockagesBlockages

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 PointsA (65) 10B (66) 8C (67) 6D (68) 4E (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)=1log10(99)=1.99999999log10(100)=2log10(999)=2.99999999log10(1000)=3log10(9999)=3.99999999log10(10000)=4log10(99999)=4.99999999log10(100000)=5log10(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.

Bone Mineral Density Math

• 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:

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:

),(exp bbsso MMNN µµ +−=

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).

bbLssLL

OL MMN

N µµ +=)ln(

bbHssHH

OH MMN

N µµ +=)ln(

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

A Pattern Recognition Problem

Linear Classifiersf x 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.

Earthwork Volume

Echocardiogram

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

Jessica Simpson

Golden Ratio: Beauty’s Secret

Silver RatioPell 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

Image Convolution

Gradient

Original, directional, Laplacian, Sharpening

Sobel and Prewitt Operators

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

Transmission in CDMA

CDMA Encoding

Signal Created by CDMA

CDMA Decoding

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 సటా్ర్ హొటల్ కు వైళళే్దిఎ0 గిలి 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?.

Minimization Problems

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?.

Tower of Honoi

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

MLM (Multi Level Marketing)

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

Atomic Blasts. What For?.

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?

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.aF = 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

UD FFF +=


m

cvmg

dt

dv

cvF

mgF

FFF

m

F

dt

dv

U

D

UD

−=

−==

+=

=

vm

cg

dt

dv −=

• 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:

( )tmcec

gmtv )/(1)( −−=

Independent variableDependent variable

ParametersForcing function


119

Analytical Solution

( )tmcec

gmtv )/(1)( −−=

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


120

)()()(

lim........)()(

1

1

01

1

i

ii

ii

tii

ii

tvm

cg

tt

tvtv

t

v

dt

dv

tt

tvtv

t

v

dt

dv

−=−−

∆∆=

−−=

∆∆≅

+

+

→∆+

+

))](([)()( 11 iiitttv

m

cgtvtv ii −−+= ++

This equation can be rearranged to yield

∆t = 2 sec

To minimize the error, use a smaller step size, ∆tNo problem, if you use a computer!

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


t (sec.) V (m/s)

0 0

2 19.60

4 32.00

8 44.82

10 47.97

12 49.96

∞ 53.39

t (sec.) V (m/s)

0 0

2 16.40

4 27.77

8 41.10

10 44.87

12 47.49

∞ 53.39

m=68.1 kg c=12.5 kg/sg=9.8 m/s

( )tmcec

gmtv )/(1)( −−= ttv

m

cgtvtv

iii∆−+=+ )]([)()( 1

∆t = 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 = 0.5 sec

t (sec.) V (m/s)

0 0

2 16.41

4 27.83

8 41.13

10 44.90

12 47.51

∞ 53.39

∆t = 0.01 sec

CONCLUSION: If you want to minimize the error, use a smaller step size, ∆t

Numerical solutionvs.

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, 2013Dear 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/AirTrackingTeacherFinal.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.

Any queries?

Thanks

Appreciationof mathematics:My observations and opinions

Engineering

Transcript of Appreciationof mathematics:My observations and opinions