Opinion
Making mistakes in mathematics
by R N A de Silva
rnades@gmail.com
“Anyone who has never made a mistake has not tried something new “, said Albert Einstein. Although mistakes are inevitable in life, learning mathematics can be extra challenging due to the pressure of having to come up with the ‘correct’ answer as it generally demands precision and accuracy. But recent neurological research indicates that making mistakes is actually good as it not only provides opportunities for learning but also contributes to the growth of the brain. Mistakes play a crucial role in the learning process and they can be considered as stepping stones on the path to mastery. Embracing mistakes will help students in building the persistence required for success in mathematics. Making mistakes will help enhance critical thinking and problem-solving skills as they prompt students to reconsider their approach, identify the source of the mistake and explore alternative solutions.
A fear of failure may hinder learning and repress creativity. It should be stressed to the students to consider mistakes as not a sign of incompetence but an opportunity for improvement. With such a mindset they are more likely to approach mathematical challenges with confidence and enthusiasm.
In 1994, researchers conducted a landmark study comparing the US education system to Japan’s education system. They found that although American teachers praised students for correct answers, they ignored incorrect responses. No discussion took place about the correct or incorrect answers. They found that the Japanese teachers had discussions about the obtained answers. Students would learn why an incorrect answer was wrong and a correct answer was right. This reflection and reinforcement would lead to much better learning. Letting students make mistakes and learn from them was found to be a key reason Japanese students outperformed Americans on global math tests.
Mistake is an example of something that does not work
Examples of situations that do not work can be just as valuable as those which work out. The Thomas Edison, who invented the lightbulb, considered the attempts that had not worked as an accomplishment of learning how it should not be done in his long journey towards the invention. His quotation “I have not failed; I have learnt 10,000 ways that won’t work” is an extremely important lesson to all of us.
According to the nature of the lesson, I sometimes make mistakes purposely when teaching students. In this way, I can figure out whether they are involved in the thinking process to the extent of being able to spot the error. It also gives satisfaction to the students, who feel that they have corrected an error made by their teacher. Further, that also helps students understand that anybody can make mistakes and they are part of the learning process.
Mistakes lead you to the correct path
Spotting an error and thereafter the thought process behind why that mistake was made may show the correct path and will help in preventing it from happening again because of the hands-on experience gained. Let us consider some examples.
(1) Square -2.
The answer is not -4. The correct method is -2 x -2 which gives +4.
The importance of the consideration of the sign of the number is shown here.
(2) Subtract 2x – 3 from x2 + 5x – 7.
The answer is not x2 + 3x -10. The correct working should be as follows:
x2 + 5x – 7 – (2x – 3) = x2 + 5x – 7 – 2x + 3 = x2 + 3x – 4.
The importance of using brackets can be seen clearly in this situation.
(3) Solve x2 = 5x.
A student giving the answer x = 5 may wonder why full marks were not awarded. The correct working should be x2 – 5x = 0 followed by x (x – 5) = 0 which gives two correct values for x. Therefore x = 0 or 5.
The cancellation of a variable is not acceptable in mathematics.
Mistakes create an opportunity for deeper understanding
Often mistakes allow students to clear misunderstandings and enhance conceptual understanding or skills-based procedures. Here are some examples.
(4) Find the square root of 25.
The answer is 5 and not +5 and -5, as the square root of a number is always positive. However, if the question was to solve an equation such as x2 = 25, then there are two correct solutions: x = +5 or x = -5.
The difference between the two has to be clearly understood.
(5) Find the square root of 94 correct to 3 significant figures.
The answer is not 9.69 because the calculated value is 9.6953.
As the number after 9 is 5 (or more) the correct answer has to be written as 9.70.
Appropriate approximation is an important concept in mathematics.
Mistakes help you to make connections with reality Mistakes may help students to focus on mathematical reasoning thereby making connections with the real world.
Consider a problem of finding the number of people in a village. The answer cannot involve fractions or decimals as you are dealing with people.
If the problem is about the annual interest rate offered by a bank, can it be as high as 50%?
If a set of numbers include numbers in the range from 1 to 10, can the mean or median be 12?
The role of the teachers and parents are of paramount importance when dealing with mistakes made by students. Here are some suggestions to make mistakes a positive experience for students.
Consider mistakes as an unavoidable and necessary experience
We all make mistakes in life. Why should it be different when learning mathematics? Mistakes happen and we can make them work to our advantage. Accept mistakes as a part of the learning process. I have come across many who hated the subject because the teachers considered them as ‘stupid’ due to some mistakes made. ‘Stupid’ is a feeling of shame and our natural reaction is to avoid its source. Instead, we should consider mistakes as an asset to the deepening of understanding a concept or a skill.
Provide timely feedback
Recognise that the earlier a problem is discovered, the easier it is to fix. Probing questions can offer students different approaches for reflecting on their thinking. Help students to overcome mistakes on their own. Students who fixes a mistake on their own experience personal success. Such an experience may lead to enhanced motivation and self-esteem and also persistence in the problem-solving process.
Analyze the mistakes and take appropriate remedial measures
The mathematical mistakes can be divided into three broad categories: careless, computational and conceptual.
Careless errors may occur due to not paying attention to details and hastiness. Some examples are misreading the question, not following directions, making mistakes with negative signs and writing wrong numbers. These can be overcome by training the students to read the question carefully and understanding what needs to be done before attempting and the cultivation of neat and orderly presentation of work.
Computational error is a mistake made with an arithmetic manipulation. When such an error occurs, all subsequent work will be affected by that error. This happens mostly due to the hastiness in arriving at the final result. Usually, such an error can be detected by checking the answer after solving. This detection will be easier if all the steps have been shown and the work presented in a logical order.
Conceptual errors occur due to the misunderstanding of the underlying concepts. Such errors are more important to be corrected than the careless and computational errors as it deals with mathematical understanding. It is an indication of a lack of necessary prerequisite knowledge to solve the problem. When this happens, foundational gaps need to be identified and appropriately fixed by the teacher or a tutor, as it may need time and effort. If a teacher notices the same conceptual error from multiple students, the teacher has to go back and reteach the topic.
Mistakes made by students provide an opportunity to get a deeper understanding of the taught concept. They can be considered part of the learning process if it is examined properly. The identification and analysis of the mistakes made by students help understand their mathematical thinking.
“When one door closes, another door opens; but we so often look so long and so regretfully upon the closed door, that we do not see the ones which open to us” said Alexander Graham Bell.
The author is an educational consultant at the Overseas School of Colombo and a senior examiner for mathematics at the International Baccalaureate Organization, UK.
Opinion
The policy of Sinhala Only and downgrading of English
In 1956 a Sri Lankan politician riding a great surge of populism, made a move that, at a stroke, disabled a functioning civil society operating in the English language medium in Sri Lanka. He had thrown the baby out with the bathwater.
It was done to huge, ecstatic public joy and applause at the time but in truth, this action had serious ramifications for the country, the effects have, no doubt, been endlessly mulled over ever since.
However, there is one effect/ aspect that cannot be easily dismissed – the use of legal English of an exact technical quality used for dispensing Jurisprudence (certainty and rational thought). These court certified decisions engendered confidence in law, investment and business not only here but most importantly, among the international business community.
Well qualified, rational men, Judges, thought rationally and impartially through all the aspects of a case in Law brought before them. They were expert in the use of this specialised English, with all its meanings and technicalities – but now, a type of concise English hardly understandable to the casual layman who may casually look through some court proceedings of yesteryear.
They made clear and precise rulings on matters of Sri Lankan Law. These were guiding principles for administrative practice. This body of case law knowledge has been built up over the years before Independence. This was in fact, something extremely valuable for business and everyday life. It brought confidence and trust – essential for conducting business.
English had been developed into a precise tool for analysing and understanding a problem, a matter, or a transaction. Words can have specific meanings, they were not, merely, the play- thing of those producing “fake news”. English words as used at that time, had meaning – they carried weight and meaning – the weight of the law!
Now many progressive countries around the world are embracing English for good economic and cultural reasons, but in complete contrast little Sri Lanka has gone into reverse!
A minority of the Sinhalese population, (the educated ones!) could immediately see at the time the problems that could arise by this move to down-grade English including its high-quality legal determinations. Unfortunately, seemingly, with the downgrading of English came a downgrading of the quality of inter- personal transactions.
A second failure was the failure to improve the “have nots” of the villagers by education. Knowledge and information can be considered a universal right. Leonard Woolf’s book “A village in the Jungle” makes use of this difference in education to prove a point. It makes infinitely good politics to reduce this education gap by education policies that rectify this important disadvantage normal people of Sri Lanka have.
But the yearning of educators to upgrade the education system as a whole, still remains a distant goal. Advanced English spoken language is encouraged individually but not at a state level. It has become an orphaned child. It is the elites that can read the standard classics such as Treasure Island or Sherlock Holmes and enjoy them.
But, perhaps now, with the country in the doldrums, more people will come to reflect on these failures of foresight and policy implementation. Isn’t the doldrums all the proof you need?
by Priyantha Hettige
Opinion
GOODBYE, DEAR SIR
It is with deep gratitude and profound sorrow that we remember Mr. K. L. F. Wijedasa, remarkable athletics coach whose influence reached far beyond the track. He passed away on November 4, exactly six months after his 93rd birthday, having led an exemplary and disciplined life that enabled him to enjoy such a long and meaningful innings. To those he trained, he was not only a masterful coach but a mentor, a friend, a steady father figure, and an enduring source of inspiration. His wisdom, kindness, and unwavering belief in every young athlete shaped countless lives, leaving a legacy that will continue to echo in the hearts of all who were fortunate enough to be guided by him.
I was privileged to be one of the many athletes who trained under his watchful eye from the time Mr. Wijedasa began his close association with Royal College in 1974. He was largely responsible for the golden era of athletics at Royal College from 1973 to 1980. In all but one of those years, Royal swept the board at all the leading Track & Field Championships — from the Senior and Junior Tarbat Shields to the Daily News Trophy Relay Carnival. Not only did the school dominate competitions, but it also produced star-class athletes such as sprinter Royce Koelmeyer; sprint and long & triple jump champions Godfrey Fernando and Ravi Waidyalankara; high jumper and pole vaulter Cletus Dep; Olympic 400m runner Chrisantha Ferdinando; sprinters Roshan Fernando and the Indraratne twins, Asela and Athula; and record-breaking high jumper Dr. Dharshana Wijegunasinghe, to name just a few.
Royal had won the Senior & Junior Tarbats as well as the Relay Carnival in 1973 by a whisker and was looking for a top-class coach to mould an exceptionally talented group of athletes for 1974 and beyond. This was when Mr. Wijedasa entered the scene, beginning a lifelong relationship with the athletes of Royal College from 1974 to 1987. He received excellent support from the then Principal, late Mr. L. D. H. Pieris; Vice Principal, late Mr. E. C. Gunesekera; and Masters-in-Charge Mr. Dharmasena, Mr. M. D. R. Senanayake, and Mr. V. A. B. Samarakone, with whom he maintained a strong and respectful rapport throughout his tenure.
An old boy of several schools — beginning at Kandegoda Sinhala Mixed School in his hometown, moving on to Dharmasoka Vidyalaya, Ambalangoda, Moratu Vidyalaya, and finally Ananda College — he excelled in both sports and studies. He later graduated in Geography, from the University of Peradeniya. During his undergraduate days, he distinguished himself as a sprinter, establishing a new National Record in the 100 metres in 1955. Beyond academics and sports, Mr. Wijedasa also demonstrated remarkable talent in drama.
Though proudly an Anandian, he became equally a Royalist through his deep association with Royal’s athletics from the 1970s. So strong was this bond that he eventually admitted his only son, Duminda, to Royal College. The hallmark of Mr. Wijedasa was his tireless dedication and immense patience as a mentor. Endurance and power training were among his strengths —disciplines that stood many of us in good stead long after we left school.
More than champions on the track, it is the individuals we became in later life that bear true testimony to his loving guidance. Such was his simplicity and warmth that we could visit him and his beloved wife, Ransiri, without appointment. Even long after our school days, we remained in close touch. Those living overseas never failed to visit him whenever they returned to Sri Lanka. These visits were filled with fond reminiscences of our sporting days, discussions on world affairs, and joyful moments of singing old Sinhala songs that he treasured.
It was only fitting, therefore, that on his last birthday on May 4 this year, the Old Royalists’ Athletic Club (ORAC) honoured him with a biography highlighting his immense contribution to athletics at Royal. I was deeply privileged to co-author this book together with Asoka Rodrigo, another old boy of the school.
Royal, however, was not the first school he coached. After joining the tutorial staff of his alma mater following graduation, he naturally coached Ananda College before moving on to Holy Family Convent, Bambalapitiya — where he first met the “love of his life,” Ransiri, a gifted and versatile sportswoman. She was not only a national champion in athletics but also a top netballer and basketball player in the 1960s. After his long and illustrious stint at Royal College, he went on to coach at schools such as Visakha Vidyalaya and Belvoir International.
The school arena was not his only forte. Mr. Wijedasa also produced several top national athletes, including D. K. Podimahattaya, Vijitha Wijesekera, Lionel Karunasena, Ransiri Serasinghe, Kosala Sahabandu, Gregory de Silva, Sunil Gunawardena, Prasad Perera, K. G. Badra, Surangani de Silva, Nandika de Silva, Chrisantha Ferdinando, Tamara Padmini, and Anula Costa. Apart from coaching, he was an efficient administrator as Director of Physical Education at the University of Colombo and held several senior positions in national sporting bodies. He served as President of the Amateur Athletic Association of Sri Lanka in 1994 and was also a founder and later President of the Ceylonese Track & Field Club. He served with distinction as a national selector, starter, judge, and highly qualified timekeeper.
The crowning joy of his life was seeing his legacy continue through his children and grandchildren. His son, Duminda, was a prominent athlete at Royal and later a National Squash player in the 1990s. In his later years, Mr. Wijedasa took great pride in seeing his granddaughter, Tejani, become a reputed throwing champion at Bishop’s College, where she currently serves as Games Captain. Her younger brother, too, is a promising athlete.
He is survived by his beloved wife, Ransiri, with whom he shared 57 years of a happy and devoted marriage, and by their two children, Duminda and Puranya. Duminda, married to Debbie, resides in Brisbane, Australia, with their two daughters, Deandra and Tennille. Puranya, married to Ruvindu, is blessed with three children — Madhuke, Tejani, and Dharishta.
Though he has left this world, the values he instilled, the lives he shaped, and the spirit he ignited on countless tracks and fields will live on forever — etched in the hearts of generations who were privileged to call him Sir (Coach).
NIRAJ DE MEL, Athletics Captain of Royal College 1976
Deputy Chairman, Old Royalists’ Athletics Club (ORAC)
Opinion
Why Sri Lanka needs a National Budget Performance and Evaluation Office
Sri Lanka is now grappling with the aftermath of the one of the gravest natural disasters in recent memory, as Cyclone Ditwah and the associated weather system continue to bring relentless rain, flash floods, and landslides across the country.
In view of the severe disaster situation, Speaker Jagath Wickramaratne had to amend the schedule for the Committee Stage debates on Budget 2026, which was subsequently passed by Parliament. There have been various interpretations of Budget 2026 by economists, the business community, academics, and civil society. Some analyses draw on economic expertise, others reflect social understanding, while certain groups read the budget through political ideology. But with the country now trying to manage a humanitarian and economic emergency, it is clear that fragmented interpretations will not suffice. This is a moment when Sri Lanka needs a unified, responsible, and collective “national reading” of the budget—one that rises above personal or political positions and focuses on safeguarding citizens, restoring stability, and guiding the nation toward recovery.
Budget 2026 is unique for several reasons. To understand it properly, we must “read” it through the lens of Sri Lanka’s current economic realities as well as the fiscal consolidation pathway outlined under the International Monetary Fund programme. Some argue that this Budget reflects a liberal policy orientation, citing several key allocations that support this view: strong investment in human capital, an infrastructure-led growth strategy, targeted support for private enterprise and MSMEs, and an emphasis on fiscal discipline and transparency.
Anyway, it can be argued that it is still too early to categorise the 2026 budget as a fully liberal budget approach, especially when considering the structural realities that continue to shape Sri Lanka’s economy. Still some sectors in Sri Lanka restricted private-sector space, with state dominance. And also, we can witness a weak performance-based management system with no strong KPI-linked monitoring or institutional performance cells. Moreover, the country still maintains a broad subsidy orientation, where extensive welfare transfers may constrain productivity unless they shift toward targeted and time-bound mechanisms. Even though we can see improved tax administration in the recent past, there is a need to have proper tax rationalisation, requiring significant simplification to become broad-based and globally competitive. These factors collectively indicate that, despite certain reform signals, it may be premature to label Budget 2026 as fully liberal in nature.
Overall, Sri Lanka needs to have proper monitoring mechanisms for the budget. Even if it is a liberal type, development, or any type of budget, we need to see how we can have a budget monitoring system.
Establishing a National Budget Performance and Evaluation Office
Whatever the budgets presented during the last seven decades, the implementation of budget proposals can always be mostly considered as around 30-50 %. Sri Lanka needs to have proper budget monitoring mechanisms. This is not only important for the budget but also for all other activities in Sri Lanka. Most of the countries in the world have this, and we can learn many best practices from them.
Establishing a National Budget Performance and Evaluation Office is essential for strengthening Sri Lanka’s fiscal governance and ensuring that public spending delivers measurable value. Such an office would provide an independent, data-driven mechanism to track budget implementation, monitor programme outcomes, and evaluate whether ministries achieve their intended results. Drawing from global best practices—including India’s PFMS-enabled monitoring and OECD programme-based budgeting frameworks—the office would develop clear KPIs, performance scorecards, and annual evaluation reports linked to national priorities. By integrating financial data, output metrics, and policy outcomes, this institution would enable evidence-based decision-making, improve budget credibility, reduce wastage, and foster greater transparency and accountability across the public sector. Ultimately, this would help shift Sri Lanka’s budgeting process from input-focused allocations toward performance-oriented results.
There is an urgent need for a paradigm shift in Sri Lanka’s economy, where export diversification, strengthened governance, and institutional efficiency become essential pillars of reform. Establishing a National Budget Performance and Evaluation Office is a critical step that can help the country address many long-standing challenges related to governance, fiscal discipline, and evidence-based decision-making. Such an institution would create the mechanisms required for transparency, accountability, and performance-focused budgeting. Ultimately, for Sri Lanka to gain greater global recognition and move toward a more stable, credible economic future, every stakeholder must be equipped with the right knowledge, tools, and systems that support disciplined financial management and a respected national identity.
by Prof. Nalin Abeysekera ✍️
-
Features4 days agoFinally, Mahinda Yapa sets the record straight
-
News6 days agoOver 35,000 drug offenders nabbed in 36 days
-
News5 days agoCyclone Ditwah leaves Sri Lanka’s biodiversity in ruins: Top scientist warns of unseen ecological disaster
-
News6 days agoRising water level in Malwathu Oya triggers alert in Thanthirimale
-
Features4 days agoHandunnetti and Colonial Shackles of English in Sri Lanka
-
Business3 days agoCabinet approves establishment of two 50 MW wind power stations in Mullikulum, Mannar region
-
Business6 days agoSri Lanka betting its tourism future on cold, hard numbers
-
News6 days agoNew landslide alerts as Ditwah aftermath worsens
