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