By Prof.Kirthi Tennakone

Decades ago, language, classics, science, and mathematics emphasiing geometry stood as the cornerstones of the high school curriculum, shaping students’ minds. These disciplines inculcate learning aptitude, creativity, abstract thinking, and empathy. Many who followed the theme in schools and colleges became professionals excelled in their art, businessmen and intellectually motivated laypeople.

In learning mathematics, geometry stands out as particularly important because the subject invigorates the mind to think deductively and imaginatively in understanding spatial relationships. Unlike in arithmetic and elementary algebra, where the problem-solving strategy proceeds with a set of operations, in geometry the student concentrates deeply looking at a sketch drawn on paper – a different kind of brain stimulating exercise.

The book Elements of Geometry by S Barnard and J M Child, widely used in Britain and our schools since the early 1900s, states geometry is the science of space and deals with shapes, sizes and positions of things. The definition agrees with the more modern view that geometry, though abstract, is essentially a study of the nature of physical space and has cosmological implications.

Virtues of learning geometry

Whatever you plan to do, geometry is invaluably relevant, directly or indirectly. Exposure to the subject influences the mindset beneficially to tackle problems beyond mathematics. So many renowned men and women have commented on the virtues of geometry.

Plato said, “Experience proves anyone who has studied geometry is infinitely quicker at grasping difficult subjects than one who has not. He attached so much importance to geometry, inscribing on the entrance to his academy the phrase “Let no one ignorant of geometry enter”.

Ibn Khaldun, 14th century Arab historian and philosopher, said “Geometry enlightens the mind and sets the mind right. All proofs are very clear and orderly and errors would not enter into geometrical reasoning. Thus, a mind that constantly applies itself to geometry is unlikely to fall into error. In this way, a person who knows geometry acquires intelligence.”

American poetess Rita Dove wrote, “I prove a theorem, house expands”.

The columnist Marlin Savant, once hailed as the world’s smartest woman, having the highest recorded IQ, wrote, “Geometry is beautifully logical, and teaches you how to think and prove things step by step. Proofs are excellent lessons in reasoning. Without logical reasoning, you are dependent on jumping to conclusions – or – worse, having empty opinions”.

The British entrepreneur and philanthropist, Dill Faulkes, describes geometry as the surest and clearest way of thinking available to us.

History of Geometry

The history of geometry tells how profoundly the discipline influenced human thinking. Geometrical ideas originated in Egypt and Babylonia as methods of measuring the extents of agricultural land. Perhaps independently in Sri Lanka as well, after initial Indian influence. Our ancient irrigation systems, monuments of rich architecture, and stupas could not be built without a practical knowledge of geometry.

Greeks looked at the subject in the spirit of abstractness, revolutionising the line of human inquiry. If they also continued to adopt geometry in the same way as Egyptians, Babylonians and Sri Lankans did, confining it only to practical uses, there wouldn’t be a modern technology.

Early Greek philosophers indulged in geometry, believing it is divine and inherent. Plato, having noted that perfect geometrical figures cannot be drawn, said they exist in a higher spiritual realm, and a man can retrieve their properties instinctively. In one of his discourses, Plato states, Socrates did an experiment to prove the point by telling an ignorant slave boy to draw a square double in area compared to one he sketched on muddy ground with a stick. The boy did it wrong in the first instance, but with a little help from Socrates, he instinctively recollected the Pythagoras theorem (both Plato and Socrates were followers of Pythagoras who lived earlier) and solved the problem. Plato’s bias to his opinion is obvious, and the experiment he attributes to Socrates may be fictitious. Nevertheless, the story shows how deep were the European philosophers, in their endeavors to fathom abstract fundamentals, paving the way for the West to dominate the world scientifically, technologically, and therefore economically.

The next bold step that enlightened geometry, radically influencing all branches of mathematics and philosophical contemplation, was the work of the Greek geometer and logician Euclid, who lived in Alexandria. He did not attribute geometry to the realm of spirituality or an inherent instinct of humans, but built its theory on the basis of a few axioms written below, taken as self-evident truths.

1. Two points are connectable by a straight line.

2. A straight-line can be extended indefinitely.

3. A circle may be drawn with any radius and an arbitrary center.

4. All right angles are equal.

5. If a straight-line intersect two other straight-lines in such a way the sum of inner angles of on one side is less than two right angles, two lines will inevitably intersect when extended in that direction.

Using the above axioms, Euclid logically deduced important properties of triangles, circles and other geometrical figures as theorems. The fifth axiom, the so-called parallel postulate, remained controversial for more than 2000 years. Mathematicians tried hard to prove it using other axioms. Finally, the impossibility of proving the assertion was understood. Many important theorems in geometry, such as the equality of the sum of three angles in a triangle to two right angles and the Pythagoras theorem, are consequences of the parallel postulate. Mathematicians in India and China knew the property of right-angled triangles attributed to Pythagoras. However, Euclid’s proof of the theorem using the parallel postulate shocked mathematicians of antiquity.

A new chapter in geometry was opened after realizing the independence of the fifth axiom. German mathematicians, Friedrich Gauss and Bernhard Riemann showed other consistent geometries exist, corresponding to figures drawn on curved surfaces. And Pythagoras Theorem is not an absolute truth but a consequence of the parallel postulate. These developments motivated Albert Einstein to formulate the general theory of relativity.

Euclid’s art of argument, making few assumptions identified as self-evident truths and logical reasoning based upon them, finds applicability and validity in affairs beyond mathematics and science. Many things you and I do depend on certain assumptions.

Examine assumptions carefully to see whether they are consistent, deduce consequences logically, and then proceed.

Abraham Lincoln, in his speeches, clearly identified assumptions, justified them as natural truths and argued logically to validate a point. After listening to a speech by Abraham Lincoln, a man asked him how he acquired such an amazing oratorical skill in presenting ideas and arguing consistently. Lincoln said, when other lawyers were sleeping and snoring, he lit a candle near the pillow and read six volumes of Euclid.

Mahatma Gandhi frequently made references to geometry in clarifying arguments. In one of his writings, Mahatma says, Euclid’s straight-line exists only in imagination, never capable of being drawn. Nevertheless, it is an important definition in geometry, yielding great results. So may a perfect bramachari exist only in imagination? But if we did not keep him constantly before the mind’s eye, we would be like a rudderless ship. The nearer the approach to the imaginary state, greater the perfection.

Teaching Geometry: Education and Science Policy Reforms

Since the time of Plato, geometry has been an integral part of academic instruction. Before Christian schools were started in the 1800s, geometry was taught only in universities. Later, these institutions demanded higher qualifications in mathematics with geometry for enrollment. Thereafter, the educationists’ world-wide emphasized formal exposure to geometry, an essential prerequisite in completing secondary level education.

Until the Education Department’s curriculum reforms were implemented in the late 1980s, Sri Lanka followed the same concept, teaching geometry as a separate subject in the 8th grade and after – largely a continuation of the school mathematics curriculum introduced by the British in the early 1900s. In those days, the Ordinary Level (OL) Mathematics, students had to sit for a separate geometry paper. Later, the geometry component in our high school mathematics syllabus was reduced, perhaps to accommodate things considered being more important in commerce and technological studies. Today, teachers and students pay less attention to geometry and concentrate on areas more straightforward in learning.

Recently, Sri Lanka, Department of Education reported that in the OL Mathematics Examination, the majority of students do not select geometry questions, and those who attempt them often give erroneous answers. Sometimes teachers advise their students to omit geometry, telling them, questions in the area are hard. Now we have a generation of mathematics teachers who neglected geometry in their school days.

The repercussions of the deficiency in teaching geometry during the past three decades have probably gone beyond OL exam performance and may account for our weaknesses in intellectual pursuits, technological innovations, and the inability to adopt an evidence-based approach in solving problems.

The poor performance in geometry can be rectified by adding more explanatory material to the OL syllabus and devoting more time to teaching. Unless the subject is made compulsory by revising the examination structure, the tendency of the teachers and students to neglect the section will continue. Furthermore, the subject should be made interesting to the students, highlighting its importance and history. Isaac Newton’s assistant has said that he witnessed the great man laugh only once when, someone asked him whether geometry has any use. Why not tell this to the students? The teachers should also tell the students, mastering geometry requires sustained mental concentration. Swami Vivekananda, a vocal advocate of the powers of concentration, said, “Just two or three days before the entrance examination, I found that I hardly knew anything of geometry. So I began to study the subject, keeping awake the whole night, and in twenty-four hours I mastered four chapters in the geometry book”.

At a time when Sri Lanka plans to propose educational reforms, to divert the human resource towards technological innovations and commercial ventures, it is prudent to note what the Russian Prime Minister, Mikhail Mishustin, said when he visited the 11th grade mathematics class in a science oriented college in Moscow 2021. Having noted that the students were attempting to answer a problem in business, he asked, “Why do you guys work on business projects in school?” Here you need to gain fundamental knowledge, and gave them a stunning problem in geometry to solve.

The message the Russian Prime Minister conveyed is clear. In schools and universities, students have to be exposed to the fundamentals to sharpen the mind and nurture creativity. With that experience, they are better equipped to specialize and deliver innovations. If fundamentals are omitted to accommodate more technological and business courses, the outcome will be counterproductive. We jump into technological fashions that emerge from time to time – biotechnology, nanotechnology, information technology and now artificial intelligence – believing they would deliver marketable products immediately. Yet the fruits of these efforts originate elsewhere, mostly in Europe and the United States of America, where schools and universities emphasize fundamental science. Teach geometry to boost the natural intelligence of our children, before embarking on artificial intelligence! For a student to enter the field of artificial intelligence and compete, he or she needs to acquire in-depth knowledge in several branches of mathematics. It is true that just like in information technology, the subject of artificial intelligence can be pursued without extra brilliance and advanced mathematical preparation. However, to make a mark and compete, those qualities are essential.

Shyness to undertake fundamental studies

The neglect of geometry is one example of our shyness to undertake intellectually challenging fundamental areas of inquiry. What the Russian Prime Minister told the mathematics class, giving a problem in geometry, is also a reminder to research institutions devoted to fundamental research. They should pursue the mandated theme without gross deviations, adulteration, or engaging in commercialization trivialities. All major innovations that pushed the West to the forefront had been curiosity driven investigations. Intellectual fantasy and dreaming and working on challenging problems, not necessarily yielding immediate results, is more important than writing papers for the purpose of getting them printed in journals.

We need policies that will qualify our students to enter ‘Plato’s Academy’.

Educational curricula and science policy reformers should keep in mind that downgrading or elimination of topics engendering qualities of abstract thinking, imagination, and empathy will lead to disastrous consequences, now beginning to be seen above Sri Lanka’s societal horizon. Bringing in reforms to accommodate technologically oriented programs curtailing the fundamentals would be ineffective. We are not competitive in technology and continue to be poor in innovations. We don’t engage in advanced frontier research, once confined to the West, but now pursued eagerly elsewhere in our region. The country doesn’t produce sufficient numbers of original thinkers, productive scientists, entrepreneurs, and knowledgeable administrators. In many situations, myth overtakes rationality, and social values are on the decline.

Our students are clever and talented. Their weakness in geometry and generating innovations is not their fault, but our wrong policies continuing for decades.We need policies that will qualify our students to enter ‘Plato’s Academy’ and our teachers and researchers to be men and women of the caliber to engage ‘there’ as philosopher mentors.

The author can be reached via email:ktenna@yahoo.co.uk

By Siri Hettige,
Emeritus Professor of Sociology,
University of Colombo

I wish to begin this article with a very broad assertion, namely, Sri Lanka’s present economic crisis is the result of a series of deliberate and short- sighted policy measures taken by post-liberalisation regimes since 1977. These policy measures led to not only structural changes in the economy but also far reaching changes in many other sectors such as education, health, transport and social welfare. As regards the economic changes, the trends have been quite clear. To understand this, one only has to follow the changes in the macroeconomic indicators over the last four decades.

If we first look at the structural changes brought about by liberalisation policies, it was quite clear that the service sector expanded rapidly, often at the expense of industrial and agricultural sectors leading to a widening trade gap as imports of industrial and even agricultural commodities increased steadily, far exceeding the value of exports. But, instead of addressing the emergent structural distortions of the economy, successive governments promoted export of labor and tourism as a way of earning foreign income to pay for rapidly increasing industrial and other goods Imported to the country.

Increasing availability of foreign exchange from worker remittances and tourism not only helped bridge the otherwise widening trade gap but also pay for all sorts of consumer goods demanded buy the increasingly affluent sections of the population. The expansion of this class was facilitated by low tax regimes maintained by successive governments. Increasing disposable incomes of a sizable segment of the population also increased the demand for private services in health, transport and education. And this led to the opening up of these sectors for private investment resulting in the proliferation of private health care providers, international schools in and around Colombo and Importation of hundreds of thousands of private vehicles.

The above developments contributed to unprecedented inequalities in the areas of health, education and passenger transport, all of which hitherto remained mostly publicly provided services. Inequality became clearly evident in all these sectors but post- liberalisation regimes failed to do anything significant to contain increasingly visible inequalities not only in household income but also the widening gap between urban and rural/estate sectors.

The failure of the post-1977 regimes to contain growing income inequality by implementing a progressive taxation policy led to decreasing state revenue, making it impossible to allocate adequate resources to publicly provided health, education and transport services. Poor quality of these services in turn created highly unequal life chances for lower income groups in society. For instance, poor educational facilities in rural and estate areas forced parents to pay for private tuition that emerged as a thriving business in all parts of the country. Poorly funded and crowded public transport services forced even many low-income people to buy transport equipment like imported motor cycles and three-wheelers to have more convenient modes of local transport, not to mention hundreds of thousands of all sorts of motor cars imported for the use of higher income groups. The same sort of development was also evident in the health sector when private provision of health care became an integral part of the health sector in Sri Lanka.

increasing cost of living as a consequence of the above developments encouraged more and more people including young men and women to migrate overseas for extended periods of employment and this helped many families to earn supplementary incomes not only to cover their day to day consumption but also to save money for children’s education, buy land, build houses, etc. But such economic gains came with considerable social costs such as the neglect of small children, break up of families and even the spread of alcohol abuse by men. Yet, increasing remittances soon became the biggest single foreign exchange earner for the country, often over 7 billion USD per year. On the other hand, increasing outflow of labor from rural and estate areas for overseas employment led to increasing costs of agricultural labor making small scale agriculture unviable, often resulting in the abandonment of many small parcels of agricultural land by farmers resulting in a decline in agriculture production and related livelihoods.

Despite social costs of labour migration, increasing worker remittances became a blessing in disguise for successive governments. In fact, populist governments began to label migrant workers as “Rata Viruwo” (“Oversees heroes”). following the equally adulatory term “Rana Viruwo” used for security service personnel fighting in the war in the north and east of the country. Availability of foreign currency earned by migrant workers enabled the governments and private companies to pay for all sorts of imports demanded by consumers, in particular those who purchased all kinds of motor cars and electrical appliances.

In spite of largely consumption driven economic growth, state revenue continued to remain low as a proportion of the GDP. In fact, state revenue declined from about 20% of GDP in the mid 1970’s to about 8% to 10% of GDP in recent years. Implications of this became so obvious when university academics asked the government to allocate 6% of the GDP for education alone. While this was obviously an impossible proposition, public investment in education had declined to about 1.5% of the GDP. In fact, this was a small fraction of what many countries, even in the Asian region invested in public education in recent decades.

The result of a very low level of public investment in education has had serious consequences for the education sector. Well to do families began to move their children from government schools to international schools that proliferated in urban areas alongside well-equipped private schools. Poorer families had no choice but rely on poorly endowed schools for their children’s education. In short, providing equal opportunities to all children and youth became an impossibility within a highly unequal education system. The situation in the health and transport sectors has not been any better than in education.

As it is evident from what is outlined above, the economic and social conditions that emerged following the implementation of neoliberal policies over the last several decades have not been equitable, just or sustainable. In fact, the conditions became worse over the last two decades when the populist regimes that came to power did not seem to care about the emerging vulnerabilities of the Sri Lankan economy due to its serious structural distortions and weaknesses. Moreover, when the public funds raised through commercial borrowings were diverted into infrastructure projects that often did not have any prospect of generating an economic return, public debts became a very serious issue that needed urgent attention. Yet, what followed was even worse when authorities began to rely on commercial borrowings to raise public funds to support government expenditure and this eventually led to high inflation imposing a heavy burden on lower income groups in the country.

The developments outlined above eventually prepared the ground for the unprecedented economic crisis when the foreign debts accumulated over several decades could no longer be serviced, resulting in the declaration of bankruptcy in early 2022.

Based on the above discussion, it can be concluded that the path to the present economic crisis was laid by shortsighted policies adopted by successive governments with callous disregard for the serious adverse effects of such policies on a large majority of people. But, what is equally important to note is that there are no political leaders and others to take responsibility for the obvious policy failures. On the other hand, the country cannot move forward, beyond the present crisis, unless a genuine national effort is made to not only agree on what went wrong but also come up with an alternative policy framework to guide desirable policy shifts and necessary institutional reforms at all levels.

On Tuesday March 28, The Island editor as is his way, struck the nail on its damn head and fearlessly made his point. He wrote: “The efficiency of the police is truly amazing,” and then added the damper: “the only problem being that it is selective.” This selectivity seems to be worsening. During the weekend their Brownie points with the government was secured at the expense of a lone person who involuntarily, we are sure, his bitter anger overcoming him, dared boo at the passing Minister Bandula Gunawardana. The many khaki Johnnies escorting the said Minister flashed into action, chased that poor guy and arrested him.

We remember the little girl who, to continue her schooling and save herself from being exposed and taunted as poor, stole three coconuts to sell to get the money she had to take to school; her sensitivity realising her mother was too poor to give her the much-needed amount. She was arrested by the police and remanded in their custody. Fortunately, word got around and the girl was rescued by someone with clout intervening.

This super efficiency in the face of murders being committed under their very noses, who knows with whose help, a harmless sportsman Thajudeen was tortured and then killed; trussed up in the seat adjoining the driver’s seat in his car pushed to crash against a wall and burst into flames, burning him to cinders and all evidence. The fire did not ignite. The case came to the very end of catching the movers and murderers and givers of orders and then poof! The case evaporared as evidence had been made to disappear by, they said, certain police officer/s. Similar with the brutal killing of Lasantha W. In these cases, and many such, the police and armed forces personnel involved are not in the public’s memory; it’s the VVIPs who are suspected of giving orders. This small fry Cassandra with a strong power of remembering may be vapourised, but the People know, remember, and may very well extract retribution since waiting for Fate or Karma to do the job takes too long.

Why on earth take notice of a boo, the tooting of a horn, the throwing of a rotten egg or overripe tomato? Such voice users and missile wielders should be thanked since much worse could be shouted out, or thrown. The patience of the masses is most often limitless; justified searing anger and galling resentment are held in check. Politicians should be thankful for this forbearance of the general public.

Across the Palk Strait

Similar to this is an event that unfolded recently in India. Resembles somewhat what happened to Ranjan Ramanayake.Poor Rahul Gandhi, MP and leader of the Congress Party and perchance a future PM of the subcontinent, has been served a two-year term of imprisonment. His crime, which one would think serious, is merely voicing a single sentence which could be taken as harmless, heard now forgotten the next moment. But no, on orders from above, the sentence he proclaimed in 2019, yes as long ago as that, said at a campaign gathering has come home to roost on orders from high up for sure. However, one wonders whether it is the police who are so perturbed with the target of the insult, unconcerned. Maybe India’s security police are also selectively over- efficient as ours is. Gandhi is accused of saying that those with the name Modi are thieves. Heinous? Not at all! Slanderous? No! Defamatory? Could be but also may not be so classified. But his saying it has brought PM Modi to the picture and over there too, it seems to be a case of pleasing, sycophantic loyalty etc.

Gandhi is given time to appeal and may go free or may, if incarcerated, gain sympathy votes for his party. He will not be able to contest the forthcoming Congress leadership election nor national elections. This last mentioned in an article Cass read means that the Lok Sabha in New Delhi does not allow those accused of crimes to enter its portals. So different over here. How many convicted of serious bribe taking, corruption, stealing, drug dealing and even rape and murder are our MPs in the House by the Diyawanne, and living off the little fat left in the land.

No to interference with justice system

Israel is in spasms of mass uprisings against the judicial reforms proposed by the government of recently re-elected PM Benjamin Netanyahu. The Star of David flag waving protests started on January 7 in Tel Aviv, spread to various locations and are masses now. The newly-appointed Justice Minister proposed judicial reforms and curtailing the power of the Supreme Court and also sought more places for govt. in the committees appointing judges. As BBC reported on Tuesday March 28, Netanyahu and his government are reconsidering the reforms.

The Defense Minister, Yoav Gallant, disagreed with the move and made known his opposition. Netanyahu promptly dismissed him which caused resignation of Israeli bigwigs like the ambassador to the US.

Cassandra has a purpose in bringing this piece of world news into her chat this Friday. Netanyahu is not the whitest of politicians, not at all. So grey and even black are many of our leaders, stained with crimes of amassing wealth and also eliminating foes and challengers to them. The Israelis attempted interfering with the judiciary and wanting more say in matters judicial. So similar to over here. Remember Chief Justice Dr, Shirani Bandranaike and how she was demeaned and grossly insulted in the Parliament premises by Rajapaksa stooges who still wield power and pontificate endlessly. Recently, wasn’t there a move to summon SC Judges to Parliament? For questioning? Attorney-at –law Prez Ranil W was the mover of this plan, his hand probably puppet-stringed. It could also very well be that he decided on his own. Attorneys at law have been protesting.

Dissimilarities appear in the matter in Israel and how things pertain in SL. They are thinking twice about the reforms and taking due note of protests. Over here strong-arm tactics and the PTA are used. Seen on TV was containing the Israeli protestors by the police with mild water cannoning and no mass temporary blinding and chocking of people, unlike in this paradise gone rotten by the hand of politicians and their vassals. The tear gas used here is not to just temporarily affect the eyes but to harm eyes and nose, lungs and life itself. And we pride ourselves as such a pacifist, democratic country!

Short take

The Island editor on Wednesday March 29 reminded his readership that ex-Prez M Sirisena is still hopeful and awaiting answers to his call for help in paying the 100 m fine imposed on him for negligence in preventing the Easter Sunday bombings and mass loss of life and serious injury. MS aka Aiyo Sirisena sure is presumptuously optimistic, stupid and dull-witted to think any Sri Lankan will contribute to save him from imprisonment. He sure must be having plenty lucre as almost all our dubious politicians have amassed. If he was scrupulously honest and has no money to spare, his brother Dudley can bail him out many times over. People were shocked by his – MS’s – changing sides but they hoot now at his SOS and methinks, wait to see him in the place he deserves to be! Bye for now, says Cassandra!