Features

Teach geometry to sharpen mind

Published

3 years ago

2023/03/31

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

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Features

Cyclones, greed and philosophy for a new world order

Published

2 hours ago

2025/12/8

Thilina Walpola

Floods caused by Cyclone Ditwah in Sri Lanka

Further to my earlier letter titled, “Psychology of Greed and Philosophy for a New World Order” (The Island 26.11.2025) it may not be far-fetched to say that the cause of the devastating cyclones that hit Sri Lanka and Indonesia last week could be traced back to human greed. Cyclones of this magnitude are said to be unusual in the equatorial region but, according to experts, the raised sea surface temperatures created the conditions for their occurrence. This is directly due to global warming which is caused by excessive emission of Greenhouse gases due to burning of fossil fuels and other activities. These activities cannot be brought under control as the rich, greedy Western powers do not want to abide by the terms and conditions agreed upon at the Paris Agreement of 2015, as was seen at the COP30 meeting in Brazil recently. Is there hope for third world countries? This is why the Global South must develop a New World Order. For this purpose, the proposed contentment/sufficiency philosophy based on morals like dhana, seela, bhavana, may provide the necessary foundation.

Further, such a philosophy need not be parochial and isolationist. It may not be necessary to adopt systems that existed in the past that suited the times but develop a system that would be practical and also pragmatic in the context of the modern world.

It must be reiterated that without controlling the force of collective greed the present destructive socioeconomic system cannot be changed. Hence the need for a philosophy that incorporates the means of controlling greed. Dhana, seela, bhavana may suit Sri Lanka and most of the East which, as mentioned in my earlier letter, share a similar philosophical heritage. The rest of the world also may have to adopt a contentment / sufficiency philosophy with strong and effective tenets that suit their culture, to bring under control the evil of greed. If not, there is no hope for the existence of the world. Global warming will destroy it with cyclones, forest fires, droughts, floods, crop failure and famine.

Leading economists had commented on the damaging effect of greed on the economy while philosophers, ancient as well as modern, had spoken about its degenerating influence on the inborn human morals. Ancient philosophers like Plato, Aristotle, and Epicurus all spoke about greed, viewing it as a destructive force that hindered a good life. They believed greed was rooted in personal immorality and prevented individuals from achieving true happiness by focusing on endless material accumulation rather than the limited wealth needed for natural needs.

Jeffry Sachs argues that greed is a destructive force that undermines social and environmental well-being, citing it as a major driver of climate change and economic inequality, referencing the ideas of Adam Smith, John Maynard Keynes, etc. Joseph Stiglitz, a Nobel Laureate economist, has criticised neoliberal ideology in similar terms.

In my earlier letter, I have discussed how contentment / sufficiency philosophy could effectively transform the socioeconomic system to one that prioritises collective well-being and sufficiency over rampant consumerism and greed, potentially leading to more sustainable economic models.

Obviously, these changes cannot be brought about without a change of attitude, morals and commitment of the rulers and the government. This cannot be achieved without a mass movement; people must realise the need for change. Such a movement would need leadership. In this regard a critical responsibility lies with the educated middle class. It is they who must give leadership to the movement that would have the goal of getting rid of the evil of excessive greed. It is they who must educate the entire nation about the need for these changes.

The middle class would be the vanguard of change. It is the middle class that has the capacity to bring about change. It is the middle class that perform as a vibrant component of the society for political stability. It is the group which supplies political philosophy, ideology, movements, guidance and leaders for the rest of the society. The poor, who are the majority, need the political wisdom and leadership of the middle class.

Further, the middle class is the font of culture, creativity, literature, art and music. Thinkers, writers, artistes, musicians are fostered by the middle class. Cultural activity of the middle class could pervade down to the poor groups and have an effect on their cultural development as well. Similarly, education of a country depends on how educated the middle class is. It is the responsibility of the middle class to provide education to the poor people.

Most importantly, the morals of a society are imbued in the middle class and it is they who foster them. As morals are crucial in the battle against greed, the middle class assume greater credentials to spearhead the movement against greed and bring in sustainable development and growth. Contentment sufficiency philosophy, based on morals, would form the strong foundation necessary for achieving the goal of a new world order. Thus, it is seen that the middle class is eminently suitable to be the vehicle that could adopt and disseminate a contentment/ sufficiency philosophy and lead the movement against the evil neo-liberal system that is destroying the world.

The Global South, which comprises the majority of the world’s poor, may have to realise, before it is too late, that it is they who are the most vulnerable to climate change though they may not be the greatest offenders who cause it. Yet, if they are to survive, they must get together and help each other to achieve self-sufficiency in the essential needs, like food, energy and medicine. Trade must not be via exploitative and weaponised currency but by means of a barter system, based on purchase power parity (PPP). The union of these countries could be an expansion of organisations,like BRICS, ASEAN, SCO, AU, etc., which already have the trade and financial arrangements though in a rudimentary state but with great potential, if only they could sort out their bilateral issues and work towards a Global South which is neither rich nor poor but sufficient, contented and safe, a lesson to the Global North. China, India and South Africa must play the lead role in this venture. They would need the support of a strong philosophy that has the capacity to fight the evil of greed, for they cannot achieve these goals if fettered by greed. The proposed contentment / sufficient philosophy would form a strong philosophical foundation for the Global South, to unite, fight greed and develop a new world order which, above all, will make it safe for life.

by Prof. N. A. de S. Amaratunga
PHD, DSc, DLITT

Features

SINHARAJA: The Living Cathedral of Sri Lanka’s Rainforest Heritage

Published

2 hours ago

2025/12/8

Thilina Walpola

Damp and thick undergrowth

When Senior biodiversity scientist Vimukthi Weeratunga speaks of Sinharaja, his voice carries the weight of four decades spent beneath its dripping emerald canopy. To him, Sri Lanka’s last great rainforest is not merely a protected area—it is “a cathedral of life,” a sanctuary where evolution whispers through every leaf, stream and shadow.

“Sinharaja is the largest and most precious tropical rainforest we have,” Weeratunga said.

“Sixty to seventy percent of the plants and animals found here exist nowhere else on Earth. This forest is the heart of endemic biodiversity in Sri Lanka.”

A Magnet for the World’s Naturalists

Sinharaja’s allure lies not in charismatic megafauna but in the world of the small and extraordinary—tiny, jewel-toned frogs; iridescent butterflies; shy serpents; and canopy birds whose songs drift like threads of silver through the mist.

“You must walk slowly in Sinharaja,” Weeratunga smiled.

“Its beauty reveals itself only to those who are patient and observant.”

For global travellers fascinated by natural history, Sinharaja remains a top draw. Nearly 90% of nature-focused visitors to Sri Lanka place Sinharaja at the top of their itinerary, generating a deep economic pulse for surrounding communities.

A Forest Etched in History

Centuries before conservationists championed its cause, Sinharaja captured the imagination of explorers and scholars. British and Dutch botanists, venturing into the island’s interior from the 17th century onward, mapped streams, documented rare orchids, and penned some of the earliest scientific records of Sri Lanka’s natural heritage.

Smallest cat

These chronicles now form the backbone of our understanding of the island’s unique ecology.

The Great Forest War: Saving Sinharaja

But Sinharaja nearly vanished.

In the 1970s, the government—guided by a timber-driven development mindset—greenlit a Canadian-assisted logging project. Forests around Sinharaja fell first; then, the chainsaws approached the ancient core.

“There was very little scientific data to counter the felling,” Weeratunga recalled.

: Poppie’s shrub frog

: Endemic Scimitar babblers

: Blue Magpie

“But people knew instinctively this was a national treasure.”

The public responded with one of the greatest environmental uprisings in Sri Lankan history. Conservation icons Thilo Hoffmann and Neluwe Gunananda Thera led a national movement. After seven tense years, the new government of 1977 halted the project.

What followed was a scientific renaissance. Leading researchers—including Prof. Savithri Gunathilake and Prof. Nimal Gunathilaka, Prof. Sarath Kottagama, and others—descended into the depths of Sinharaja, documenting every possible facet of its biodiversity.

Thilak

“Those studies paved the way for Sinharaja to become Sri Lanka’s very first natural World Heritage Site,” Weeratunga noted proudly.

: Vimukthi

: Nadika

: Janaka

A Book Woven From 30 Years of Field Wisdom

For Weeratunga, Sinharaja is more than academic terrain—it is home. Since joining the Forest Department in 1985 as a young researcher, he has trekked, photographed, documented and celebrated its secrets.

Now, decades later, he joins Dr. Thilak Jayaratne, the late Dr. Janaka Gallangoda, and Nadika Hapuarachchi in producing, what he calls, the most comprehensive book ever written on Sinharaja.

“This will be the first major publication on Sinharaja since the early 1980s,” he said.

“It covers ecology, history, flora, fauna—and includes rare photographs taken over nearly 30 years.”

Some images were captured after weeks of waiting. Others after years—like the mysterious mass-flowering episodes where clusters of forest giants bloom in synchrony, or the delicate jewels of the understory: tiny jumping spiders, elusive amphibians, and canopy dwellers glimpsed only once in a lifetime.

The book even includes underwater photography from Sinharaja’s crystal-clear streams—worlds unseen by most visitors.

A Tribute to a Departed Friend

Halfway through the project, tragedy struck: co-author Dr. Janaka Gallangoda passed away.

“We stopped the project for a while,” Weeratunga said quietly.

“But Dr. Thilak Jayaratne reminded us that Janaka lived for this forest. So we completed the book in his memory. One of our authors now watches over Sinharaja from above.”

Jumping spide

An Invitation to the Public

A special exhibition, showcasing highlights from the book, will be held on 13–14 December, 2025, in Colombo.

“We cannot show Sinharaja in one gallery,” he laughed.

“But we can show a single drop of its beauty—enough to spark curiosity.”

A Forest That Must Endure

What makes the book special, he emphasises, is its accessibility.

“We wrote it in simple, clear language—no heavy jargon—so that everyone can understand why Sinharaja is irreplaceable,” Weeratunga said.

“If people know its value, they will protect it.”

To him, Sinharaja is more than a rainforest.

It is Sri Lanka’s living heritage.

A sanctuary of evolution.

A sacred, breathing cathedral that must endure for generations to come.

By Ifham Nizam

Features

How Knuckles was sold out

Published

2 hours ago

2025/12/8

Thilina Walpola

Knuckles range

Leaked RTI Files Reveal Conflicting Approvals, Missing Assessments, and Silent Officials

“This Was Not Mismanagement — It Was a Structured Failure”— CEJ’s Dilena Pathragoda

An investigation, backed by newly released Right to Information (RTI) files, exposes a troubling sequence of events in which multiple state agencies appear to have enabled — or quietly tolerated — unauthorised road construction inside the Knuckles Conservation Forest, a UNESCO World Heritage site.

At the centre of the unfolding scandal is a trail of contradictory letters, unexplained delays, unsigned inspection reports, and sudden reversals by key government offices.

“What these documents show is not confusion or oversight. It is a structured failure,” said Dilena Pathragoda, Executive Director of the Centre for Environmental Justice (CEJ), who has been analysing the leaked records.

“Officials knew the legal requirements. They ignored them. They knew the ecological risks. They dismissed them. The evidence points to a deliberate weakening of safeguards meant to protect one of Sri Lanka’s most fragile ecosystems.”

A Paper Trail of Contradictions

RTI disclosures obtained by activists reveal:

Approvals issued before mandatory field inspections were carried out

Three departments claiming they “did not authorise” the same section of the road

A suspiciously backdated letter clearing a segment already under construction

Internal memos flagging “missing evaluation data” that were never addressed

“No-objection” notes do not hold any legal weight for work inside protected areas, experts say.

One senior officer’s signature appears on two letters with opposing conclusions, sent just three weeks apart — a discrepancy that has raised serious questions within the conservation community.

“This is the kind of documentation that usually surfaces only after damage is done,” Pathragoda said. “It shows a chain of administrative behaviour designed to delay scrutiny until the bulldozers moved in.”

The Silence of the Agencies

Perhaps, more alarming is the behaviour of the regulatory bodies.

Multiple departments — including those legally mandated to halt unauthorised work — acknowledged concerns in internal exchanges but issued no public warnings, took no enforcement action, and allowed machinery to continue operating.

“That silence is the real red flag,” Pathragoda noted.

“Silence is rarely accidental in cases like this. Silence protects someone.”

On the Ground: Damage Already Visible

Independent field teams report:

Fresh erosion scars on steep slopes

Sediment-laden water in downstream streams

Disturbed buffer zones

Workers claiming that they were instructed to “complete the section quickly”

Satellite images from the past two months show accelerated clearing around the contested route.

Environmental experts warn that once the hydrology of the Knuckles slopes is altered, the consequences could be irreversible.

CEJ: “Name Every Official Involved”

CEJ is preparing a formal complaint demanding a multi-agency investigation.

Pathragoda insists that responsibility must be traced along the entire chain — from field officers to approving authorities.

“Every signature, every omission, every backdated approval must be examined,” she said.

“If laws were violated, then prosecutions must follow. Not warnings. Not transfers. Prosecutions.”

A Scandal Still Unfolding

More RTI documents are expected to come out next week, including internal audits and communication logs that could deepen the crisis for several agencies.

As the paper trail widens, one thing is increasingly clear: what happened in Knuckles is not an isolated act — it is an institutional failure, executed quietly, and revealed only because citizens insisted on answers.

by Ifham Nizam