GOTTFRIED WILHELM LEIBNIZ THE FATHER OF CALCULUS 372 TH BIRTHDAY

https://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz

https://www.youtube.com/watch?v=FPCzEP0oD7I

APPROACHES IN TEACHING MATHEMATICS

APPROACHES IN TEACHING MATHEMATICS

 INQUIRY TEACHING

Inquiry Teaching involves providing learners with content-related 

problems that serve as the foci for class research activities. The 

teacher provides/presents a problem then the learners identify the 

problem.

THE STEPS IN THE PROCESS OF INQUIRY

• Present discrepant event or specific problematic situation.

• Encourage observation for developing a statement of research 

   objectives

• Ask students for observation and explanation

• Encourage the testing of the hypothesis

• Develop tentative conclusion and generalization

• Debrief the process

DEMONSTRATION APPROACH 

Demonstration Approach is a teaching strategy in which the 

teacher engages “in a learning task other than just talking

about it”.

DISCOVERY APPROACH

Discovery Learning is “International Learning” . Both the teacher 

and the learner play active roles in discovery learning depending 

upon on the role that the teacher plays, this can range from guided

discovery (needs strict supervision) to free or pure discovery (very 

little supervision needed)

Steps of lesson planning were adopted as given by Carin and 

Surd  (1981)

1) Statement of the problem.

2) Previous knowledge.

3) Concept to be developed.

4) Specific objectives.

5) Teaching aids.

 6) Presentation.

7) Questions of Discussion.

8) Investigative activities of students.

 9) Observation table made by the students.

10) Generalization 

11) Open questions.

12) Teacher activity.

MATH-LAB APPROACH

The Mathematics Laboratory Approach is a method of teaching 

whereby children in small groups work through an assignment/task 

card, learn and discover mathematics for themselves.

PRACTICAL WORK APPROACH (PWA)

The learners in this approach, manipulate concrete objects and/or 

perform activities to arrive at a conceptual understanding of 

phenomena, situation, or concept. The environment is a laboratory

where the natural events/phenomena can be subjects of 

mathematical or scientific investigations.

INDIVIDUALIZED INSTRUCTION USING MODULES

The application of Individualized Instruction permits the learners to

 progress by mastering steps through the curriculum at his/her own 

rate and independently of the progress of other pupils.

BRAINSTORMING

 It is a teaching strategy in which the teacher elicits from the 

learners as many ideas as possible but refrains from evaluating 

them until all possible ideas have been generated.

BRAINSTORMING USUALLY OCCURS IN 4 PHASES

 1) problem identification,

 2) idea generation,

 3) idea evaluation, and

4) solution implementation and evaluation.

PROBLEM-SOLVING

Problem-solving can best be defined as a learner-directed strategy 

in which learners “think patiently and analytically about complex 

situations in order to find answers to questions”. A problem is 

defined as a “situation in which you are trying to reach some goal, 

and must find means for getting there”.

COOPERATIVE LEARNING

Cooperative learning is helpful in eliminating competition among 

learners. It encourages them to work together towards common 

goals. It fosters positive intergroup attitudes in the classroom.

INTEGRATIVE TECHNIQUE

The Integrated Curriculum Mode (Integrative teaching to some) is 

both a “method of teaching and a way of organizing the 

instructional program so that many subject areas and skills 

provided in the curriculum can be linked to one another”.

MAGIC HEXAGON

A magic hexagon of order n is an arrangement of numbers in a centered hexagonal pattern with n cells on each edge, in such a way that the numbers in each row, in all three directions, sum to the same magic constant. 
A normal magic hexagon contains the consecutive integers from 1 to 3n² − 3n + 1. 
It turns out that magic hexagons exist only for n = 1 (which is trivial) and n = 3.

Arsen Zahray discovered these order 4 and 5 hexagons:

The order 4 hexagon starts with 3 and ends with 39, its rows summing to 111. The order 5 hexagon starts with 6 and ends with 66 and sums to 244. An order 6 hexagon can be seen below. It was created by Louis Hoelbling, October 11th, 2004

BRAHMA GUPTA - GREAT INDIAN MATHEMATICIAN

Brahma Gupta was a great Indian mathematician of 7th century C E. He was born in a village called Billamalla in Rajasthan in the year 598 CE. He moved to Ujjain in central India which had a famous school of mathematics. In that school, his predecessors were the famous astronomers Varahamihira and Aryabhata.

In those days, astronomers did lot of mathematical work too and the distinction between astronomy and mathematics was not clear cut. Brahma Gupta was also known as a great astronomer and became head of the Ujjain observatory. 

1  Number Theory

Perhaps his greatest contribution was in number theory. He developed the use of zero with precise equations: if n is a number ,then 0 + n = 0, 0 x 0 =0 and so on.

He also enunciated that negative numbers could be used in what we now call "algebra". He found out that while taking the square root we get two roots--one positive and one negative. Thus square root of 9 is either +3 or -3.

2  Brahma Gupta Formula

For a cyclic quadrilateral --- that is a quadrilateral inscribed in a circle ---- the area A is given by :

A = square root( s-a)(s-b)(s-c)(s-d)

where a, b,c,and d are sides of the quadrilateral and s is the semi-perimeter: s= (a+b+c+d)/2

[This can be extended to non-cyclic quadrilateral also;] This reminds us of the formula for area of a triangle given by Heron [of Alexandria].

Note that BG's formula reduces to Heron's formula when d goes to zero:

Heron's formula:   Area = sqrt( s (s-a)(s-ab)(s-a))

It is a moot point whether BG wanted to extend Heron's formula or derived this independently. It is quite possible that he was aware of Heron's formula.

3  Approximation to Pi

Almost all astronomers and mathematicians have been fascinated by the irrational number pi and had approximated it in several ways. They needed the value of pi for many computations. 

Egyptians used the ratio of 256/81 = 3.1605 as pi for all calculations. 256/81 = (4x4x4x4)/(3x3x3x3)

Early Greeks used pi = 3 or  following Archimedes work, pi= 3 + 10/71 or simply, pi = 3 +1/7 or pi = 22/7 --  a ratio   often used by school students even today.

Brahma Gupta approximated pi to square root of 10 which is 3.16 . This is close to 3.14159 and was perhaps convenient to use in astronomical calculations.

{Bhaskara II used the ratio of 355/113 for pi,yielding 3.14159.]

Brahma Gupta's work was translated by Arab mathematicians into Arabic and became part of Arab math in their schools,especially the one that developed in Baghdad. 

The book "Sindhind" contained his works on number theory for Arab mathematicians. The noted    mathematician al-Khwarizmi wrote his book of Algebra in 830 CE, including BG's works. 

By 12th century, the work of Brahma Gupta was widely known in Europe.It was at this time Bhaskara wrote his further work    on Pell's equation--around 1150 CE.

It should be noted that BG's works, like works of other Indian mathematicians, were written in verse form in Sanskrit language .

Like other Indian astronomer-mathematicians,much of his work might have been motivated by astronomical problems.

BG wrote two books, the second one at the age of 69.

Mathematics Jingles

We’re Wise 

To the tune of Magtanim ay di Biro
 Learning Math is really fun 
New ideas every time 
There is joy for everyone Problem solving satisfies.
 I am glad, you are glad For your answer are all right 
We all feel that we are bright Solving problem makes us wise. 

Math Time 

To the tune of It’s a Small Word 
Oh, it’s math time after all (3x)
 Come together and come all 
There is just one class we enjoy a lot. 
Where our mind thinks hard and compute so fast 
Though the drills are so fast And the problem so tough 
We enjoy our class in Math.

What is a cryptarithm ?

The substitution of numbers for letters is called cryptography, and a cryptarithm is a mathematical problem in which letters are substituted for numbers. Here is a sample: 

The problem is to find the number ABC which has been squared. Here is how to go about solving it: Start with C which is the last digit of the number and its square. 
There are only four numbers, which, when multiplied by themselves, will have their last digit the same as the number. 
They are: 0 (0 x 0=0),1 (1 x 1=1),5 (5 x 5 = 25) and 6 (6 X 6 = 36) . C cannot be equal to 0, for when you multiply a number by zero, you get zero, but in this problem we multiply C by Band get E. 
Neither can C equal 6. 
Note in the center column of the addition, we have D + C + C = D. If C equals 6, it would not be possible to add 6 + 6 + any number and have the sum equal the missing number. C cannot be equal to 1, since C X ABC would equal ABC, but in this problem it equals DBC. 
Therefore, C must be equal to 5. We also know the number of another letter: A . We see in the multiplication that A X ABC = ABC. Therefore, A must equal 1. 
We now have two digits: A = 1 and C = 5. 
Write the problem over substituting the known numbers:

Look at the center column where D + 5 + 5 is used in the problem. We now know that 10 + D equals D and we carry 1. Therefore, we know that 1 + B + B = 5. The only number that B can stand for is 2 since 1 + 2 + 2 =5. The problem is now solved: ABC= 125. 

How can you use your watch as a compass ?

Hold your watch so that it is level with the ground and point the hour hand toward the su'n. South is halfway between the hour hand and the 12 o'clock mark. For example, at 5 minN utes after 10 A.M. (Standard Time), with the hour hand pointing at the sun, halfway between the 10 and 12 - or at the 11 - is the location of south. An imaginary line drawn through 11 and 5 points north and south.