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. 

MESOPTAMIAN MATHEMATICIANS

Who were the Mesopotamian mathematicians ? 
About a thousand miles east of the Nile lies the fertile valley of the Tigris and Euphrates, once known as Mesopotamia. During early history this land was the home of the Sumerians, Chaldeans, Assyrians and Babylonians. In some ways, their society was similar to Egypt's. Their mathematicians also belonged to the priestly caste. Unlike Egypt, Mesopotamia carried on extensive foreign trade with the people to the west in Lebanon, to the north in Asia Minor, to the east in India, and possibly even with China. What we know of their mathematics comes to us from the baked clay tablets on which they wrote. The Babylonians had advanced mathematical knowledge as far back as 2500 B.C. 
We know that they inherited from the Sumerians their cuneiform, or wedge-shape, writing and numerals. We are indebted to these people for several of our basic mathematical concepts and notations.

THE HARDY - WEINBERG LAW - MATHEMATICS AND GENETICS

This was explained by the Hardy–Weinberg law, an application of basic mathematics to genetics. It explains how, in the Mendelian theory of inheritance, a dominant gene does not take over completely and a recessive gene does not die out.
G.H. Hardy was an English mathematician who prided himself on the nonapplicability of mathematics. He was a great researcher in pure mathematics but is probably more widely known for this single contribution to genetics – which started life as a piece of mathematics on the back of an envelope done after a cricket match. Wilhelm Weinberg came from a very different background. A
general medical practitioner in Germany, he was a geneticist all his life. He discovered the law at the same time as Hardy, around 1908.
The law relates to a large population in which mating happens at random. There are no preferred pairings so that, for instance, blue-eyed people do not prefer to mate with blue-eyed people. After mating, the child receives one factor from each parent. For example, a hybrid genotype bB mating with a hybrid bB
can produce any one of bb, bB, BB, but a bb mating with a BB can only produce a hybrid bB. What is the probability of a b-factor being transmitted ? Counting the number of b-factors there are two b-factors for each bb genotype and one b factor for each bB genotype giving, as a proportion, a total of three b-factors out
of 10 (in our example of a population with 1:1:3 proportions of the three genotypes). The transmission probability of a b-factor being included in the genotype of a child is therefore 3/10 or 0.3. The transmission probability of a B factor being included is 7/10 or 0.7. The probability of the genotype bb being included in the next generation, for example, is therefore 0.3 × 0.3 = 0.09. The complete set of probabilities is summarized in the table.

The hybrid genotypes bB and Bb are identical so the probability of this occurring is 0.21 + 0.21 = 0.42. Expressed as percentages, the ratios of genotypes bb, bB and BB in the new generation are 9%, 42% and 49%. Because B is the dominant factor, 42% + 49% = 91% of the first generation will have brown eyes. Only an individual with genotype bb will display the observable
characteristics of the b factor, so only 9% of the population will have blue eyes.
The initial distribution of genotypes was 20%, 20% and 60% and in the new generation the distribution of genotypes is 9%, 42% and 49%. What happens next ? Let’s see what happens if a new generation is obtained from this one by random mating. 
The proportion of b-factors is 0.09 + ½ × 0.42 = 0.3, the proportion of B-factors is ½ × 0.42 + 0.49 = 0.7. These are identical to the
previous transmission probabilities of the factors b and B. The distribution of genotypes bb, bB and BB in the further generation is therefore the same as for the previous generation, and in particular the genotype bb which gives blue eyes does not die out but remains stable at 9% of the population. Successive proportions of genotypes during a sequence of random matings are therefore
20%, 20%, 60% → 9%, 42%, 49% → . . . → 9%, 42%, 49%
This is in accordance with the Hardy–Weinberg law: after one generation the genotype proportions remain constant from generation to generation, and the transmission probabilities are constant too.
Hardy’s argument
To see that the Hardy–Weinberg law works for any initial population, not just the 20%, 20% and 60% one that we selected in our example, we can do no better than refer to Hardy’s own argument which he wrote to the editor of the American journal Science in 1908.
Hardy begins with the initial distribution of genotypes bb, bB and BB as p, 2r and q and the transmission probabilities p + r and r + q. In our numerical example (of 20%, 20%, 60%), p = 0.2, 2r = 0.2 and q = 0.6. The transmission probabilities of the factors b and B are p + r = 0.2 + 0.1 = 0.3 and r + q = 0.1 + 0.6 = 0.7. 
What if there were a different initial distribution of the genotypes
bb, bB and BB and we started with, say, 10%, 60% and 30%? How would the Hardy–Weinberg law work in this case ? Here we would have p = 0.1, 2r = 0.6 an d q = 0.3 and the transmission Probabilities of the factors b and B are respectively p + r = 0.4 and r + q = 0.6. So the distribution of next generation of genotypes is 16%, 48% and 36%. Successive proportions of the genotypes
bb, bB, and BB after random matings are and the proportions settles down after one generation, as before, and the transmission probabilities of 0.4 and 0.6 remain constant. With these figures 16% of the population will have blue eyes and 48% + 36% = 84% will have brown eyes because B is dominant in the genotype bB.
10%, 60%, 30% → 16%, 48%, 36% → . . . → 16%, 48%, 36%
So the Hardy–Weinberg law implies that these proportions of genotypes bb, bB and BB will remain constant from generation to generation whatever the initial distribution of factors in the population. The dominant B gene does not take over and the proportions of genotypes are intrinsically stable.
Hardy stressed that his model was only approximate. Its simplicity and elegance depended on many assumptions which do not hold in real life. In the model the probability of gene mutation or changes in the genes themselves has been discounted, and the consequence of the transmission proportions being constant means it has nothing to say about evolution. In real life there is ‘genetic drift’ and the transmission probabilities of the factors do not stay constant. This
will cause variations in the overall proportions and new species will evolve.
The Hardy–Weinberg law drew together Mendel’s theory – the ‘quantum theory’ of genetics – and Darwinism and natural selection in an intrinsic way. It awaited the genius of R.A. Fisher to reconcile the Mendelian theory of inheritance with the continuous theory where characteristics evolve.
What was missing in the science of genetics until the 1950s was a physical understanding of the genetic material itself. Then there was a dramatic advance contributed by Francis Crick, James Watson, Maurice Wilkins and Rosalind Franklin. The medium was deoxyribonucleic acid or DNA. Mathematics is needed to model the famous double helix (or a pair of spirals wrapped around a
cylinder). The genes are located on segments of this double helix.
Mathematics is indispensable in studying genetics. From the basic geometry of the spirals of DNA and the potentially sophisticated Hardy–Weinberg law, mathematical models dealing with many characteristics (not just eye-colour) including male–female differences and also non-random mating have been developed. The science of genetics has also repaid the compliment to mathematics by suggesting new branches of abstract algebra of interest for their intriguing mathematical properties.

THE FANO PLANE

The Fano plane geometry was discovered at about the same time as Pick’s formula, but has nothing to do with measuring anything at all. Named after the Italian mathematician Gino Fano, who pioneered the study of finite geometry, the Fano plane is the simplest example of a ‘projective’ geometry. It has only seven
points and seven lines.

The seven points are labelled A, B, C, D, E, F and G. It is easy to pick out six of the seven lines but where is the seventh? The properties of the geometry and the way the diagram is constructed make it necessary to treat the seventh line as DFG – the circle passing through D, F and G. This is no problem since lines in
discrete geometry do not have to be ‘straight’ in the conventional sense.
This little geometry has many properties, for example:

• every pair of points determines one line passing through both,

• every pair of lines determines one point lying on both.

These two properties illustrate the remarkable duality which occurs in geometries of this kind. The second property is just the first with the words ‘point’ and ‘line’ swapped over, and likewise the first is just the second with the same swaps.
If, in any true statement, we swap the two words and make small adjustments to correct the language, we get another true statement. Projective geometry is very symmetrical. Not so Euclidean geometry. In Euclidean geometry there are parallel lines, that is pairs of lines which never meet. We can quite happily speak
of the concept of parallelism in Euclidean geometry. This is not possible in projective geometry. In projective geometry all pairs of lines meet in a point. For mathematicians this means Euclidean geometry is an inferior sort of geometry.

If we remove one line and its points from the Fano plane we are once more back in the realm of unsymmetrical Euclidean geometry and the existence of parallel lines. Suppose we remove the ‘circular’ line DFG to give a Euclidean diagram.
With one line fewer there are now six lines: AB, AC, AE, BC, BE and CE. There are now pairs of lines which are ‘parallel’, namely AB and CE, AC and BE, and BC and AE. Lines are parallel in this sense if they have no points in common – like the lines AB and CE.

The Fano plane occupies an iconic position in mathematics because of its connection to so many ideas and applications. It is one key to Thomas Kirkman’s schoolgirl problem . 
In the theory of designing experiments the Fano plane appears as a protean example, a Steiner Triple System (STS). Given a finite number of n objects an STS is a way of dividing them into blocks of three so that every pair taken from the n objects is in exactly one block. Given the seven objects A, B, C, D, E, F and G the blocks in the STS correspond to the lines of the Fano plane.