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Opinion

Making mistakes in mathematics

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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.



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Opinion

Future of world religions

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I refer to Dr Upul Wijayawardena’s recent letter on science and religion. I agree with him on the fate of religions in the modern world and would like to add the following observations on the evolution of religion and the advent of AI.

Religious philosophy evolved only in ancient India while it remained static with monotheism in other parts of the world. When the polytheistic Vedic religion arrived with Aryan migration to India, they found the ancient version of Jainism, which may have been practised over four thousand years as evidenced by some figures in meditation posture found in Mohenjo daro. By that time the indigenous religion had already abandoned the creator god or gods but still believed the mind as a permanent entity athman that survived death and continued in a cycle of reincarnation (samsara) due to results of any harmful action (kamma) even unintended performed in life collected in the athman. Salvation to a happy all-knowing permanent state named Kevala could be attained by the athman with extreme degree of self-mortification to destroy the past kamma attached to it, refraining from new kamma and developing the mind by meditation. As observed in progression from Veda to Upanishads the Aryan religion, evolution progressed with adoption of kamma, the cyclic existence of samsara, meditation and ethical behaviour in addition to prayer and sacrifice they practised for moksha.

The Buddha appeared in this philosophical atmosphere and denied both creator god and a permanent entity of athman. His psychology describes human behaviour based on the mind as a stream of changing consciousness. However, he accepted the cyclic existence of samsara; he said he verified the survival of consciousness at death with help of knowledge he gained from meditation. He prescribed a middle path to attain nirvana, a permanent state of happiness, avoiding self-mortification and redefined kamma to intentional activity only, both ethical and unethical resulting in bad and good results. He disagreed with the idea of kamma as the cause of rebirth. Following the Buddha, Vedantha religion reformed god to universal athman or brahman and evolved to modern Hinduism with a mixture of polytheism.

In his description of living beings in the universe, the Buddha has described some with a body made of fine material (brahma beings) in comparison to the gross material body of human and animal beings. One may assume the brahma body may not be carbon based.

The Buddha has also described living beings with physical body only (asannasatta) and another variety with mind only without a body (arupa satta). Therefore, it is not surprising to Buddhists that AI may evolve to become living beings if it develops a readily and freely available mobile source of energy and takes over the world with its superior capacity. AI already has information about Buddhist psychology. With that knowledge ‘AI beings’ may even become Buddhists not given to rituals.

According to the Buddha, brahma beings and asannasatta beings have very long lifespans as their bodies are not subject to decay and illness human beings develop. The new AI beings also may live as long as they wish because they may be able to repair themselves in case of developing physical defects. They would need only ethical behaviour to avoid destroying themselves by peaceful existence if they took over the world, as feared in some quarters.

However, as long as human beings exist in the world with death and uncertainty of what happens after death, they need a religion and rituals to give them psychological support. Even those who profess death is the end of life may have some doubt troubling their mind from time to time and may think of religion in times of adversity.

Ideally, human beings are free to accept one of the many versions of religious philosophies and respect each other’s choice and live in harmony. Unfortunately, extreme degrees of religious versions that cause division, unrest and violence will always continue to exist together with religions in the world.

Upali Abeysiri

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Opinion

The Cream of the Earth: Why Sri Lanka needs a Fruit Consumers Movement

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By Lalin I De Silva

Sri Lanka speaks constantly about hospitals, medicines, food imports, agricultural productivity and the rising cost of living. Yet there is one simple question we rarely ask:

Are we eating enough of the food that our own soil can naturally give us?

Fruit is perhaps one of the finest gifts of the earth to humankind. A ripe mango, banana, papaya, guava or avocado comes to us almost ready to eat. Sunlight, rain, soil, microorganisms, minerals and the labour of a farmer have already done most of the work.

That is why I like to describe fruit as the “cream of the earth.”

This is, of course, not an argument that fruit alone can prevent every disease, or guarantee longevity. Science does not support such exaggerated claims. But there is strong international evidence that regular consumption of fruits and vegetables forms an important part of a healthy diet and is associated with lower risks of several major non-communicable diseases.

Sri Lanka’s own dietary guidance encourages the daily consumption of fruit as part of a balanced diet.

The tragedy is that we live on an island capable of producing an extraordinary variety of fruits, while many families increasingly depend on highly processed foods, sugary drinks and imported snacks.

This deserves national attention.

We Produce — But Do We Connect?

Sri Lanka grows banana, mango, pineapple, papaya, avocado, rambutan, mangosteen, guava, woodapple, beli, pomegranate, durian and many lesser-known indigenous fruits.

Yet our fruit value chain remains fragmented.

The farmer often does not know the final consumer.

The consumer rarely knows the farmer.

Fruit travels through several intermediaries, sometimes suffering poor handling, unnecessary damage and price escalation along the way.

Consumers may not know where a fruit was grown, when it was harvested, whether it was naturally matured or what crop-protection practices were followed.

Meanwhile, a careful farmer, producing exceptional fruit, may receive almost the same farm-gate price as someone producing an inferior product.

That is not value-chain management.

It is precisely this disconnect that a new citizen-led initiative could begin to address.

25 Districts — 25 Consumer Societies

A proposal now emerging is to establish a Pure Sri Lankan Authentic Healthy Fruit Consumers Society in each of Sri Lanka’s 25 districts.

The concept should not become another organisation full of titles, meetings and ceremonial functions.

Its purpose must be practical.

Each district society could bring together consumers, growers, doctors, nutritionists, horticulturists, schools, universities, retailers, hotels, processors and young entrepreneurs.

The first task would be to understand the district itself.

What fruits are grown?

Where are they grown?

When are they available?

Which traditional varieties are disappearing?

Where can consumers purchase genuinely good fruits?

Where are the post-harvest losses?

What do farmers need?

What do consumers actually want?

From this information, each district could eventually develop its own Fruit Map.

Twenty-five district maps could then become a National Fruit Map of Sri Lanka.

From Anonymous Fruit to Traceable Fruit

Imagine buying a mango and scanning a QR code.

You could see the district, grower, variety, approximate harvest date and production information.

You could perhaps even provide feedback on taste.

That may sound futuristic, but the technology already exists.

The problem is not technology.

The problem is that our food system was never designed to connect the person who grows the fruit with the person who eats it.

Traceability could change that.

The principle is simple:

Know your farmer. Know your fruit.

Let the Consumer Lead

For decades, agriculture has largely been driven from the production side.

“How much can we produce?” has been the dominant question.

A healthier agricultural economy should begin with another question:

What does the consumer need, value and trust?

If thousands of consumers deliberately demand fresh, safe, flavorful and traceable Sri Lankan fruit, the message travels backwards through the entire value chain.

Retailers respond.

Farmers respond.

Nurseries respond.

Researchers respond.

Investors respond.

Government eventually has to respond.

That is the real power of an organised consumer movement.

Start With Children

Perhaps the most powerful campaign message could be remarkably simple:

Two Fruits a Day.

Imagine a Sri Lankan child coming home from school and asking:

“What are our fruits today?”

Imagine every school having a small fruit garden.

Imagine children being able to identify 10 Sri Lankan fruits, understand their seasons and know where they are grown.

Such behavioral change cannot be achieved merely by issuing circulars from Colombo.

It must become part of family culture.

Kandy Could Begin

Kandy would be a logical district in which to pilot the concept.

Its climatic diversity, surrounding agricultural communities, educational institutions, hotels, markets and strong plantation heritage provide an excellent testing ground.

A successful Kandy model could then be adapted—not blindly copied—across the other 24 districts.

The objective should ultimately be much larger than selling more fruit.

It is about reconnecting soil, farmer, food, consumer and health.

Sri Lanka does not need to invent every solution from outside.

Sometimes the answer may already be hanging from a tree beside us.

We simply have to recognise its value.

Know Your Farmer. Know Your Fruit. Live Well.

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Opinion

‘Outpost of Empire’

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Review

by Colleen Figg, Editor
and Reviewer – Amazon, New Voices Publishing

“Fernando echoes Macbeth with the line, “There is judgment here.” It captures the question running through the novel: can anyone participate in corruption and remain untouched by it? Beneath this satire is a dark story about power and the danger of believing, that someone else will always pay the price.”

Shavindra Fernando’s Outpost of Empire is a political novel set in the fictional country of Lackadasia. At its heart is a strong idea: an exploration of world politics, terrorism and the ways in which powerful countries manipulate smaller ones. Fernando is clearly writing about political issues that matter deeply to him.

The names immediately establish the book’s satirical character. Lackadasia, Burgherbugaria and the various Klowns suggest a world in which the people and places are exaggerated, but the issues beneath the exaggeration are recognisable. The playful language allows Fernando to approach terrorism, religious conflict and political corruption from an unusual angle.

Henry Stockwell is a central character: an ambassador who arrives in Lackadasia presenting himself as a friend of the country. He believes he understands how power works and treats the country, its politicians and its people as pawns on his personal chessboard. In his arrogance, he never doubts that he controls events rather than being controlled by them. This confidence makes him a compelling figure and in him, we explore the workings of political influence and the potential traps contained therein.

As the story develops, Fernando reveals a political world where causes are adopted when they are useful, people are exploited and discarded, and the truth becomes whatever those in power need it to be. Ordinary people suffer the consequences of decisions made far beyond their reach or knowledge, while those making those decisions continue to justify their actions. Even the people who help keep this world running may not be as important or as safe as they imagine.

Stockwell brings this idea into focus. He recognises corruption in others and willingly takes part in it himself, yet he cannot see that he might also suffer its consequences. Ultimately, he thinks his position will keep him safe. He is intelligent enough to understand the game, but not insightful enough to grasp the limits of his place within it. This blindness gives the novel a strong sense of irony.

The satirical approach gives Fernando room to criticise political hypocrisy without turning the novel into a direct political argument. By using fictional countries and exaggerated names, he creates some distance from specific governments and events while still making the larger targets of the book clear. The humour does not take away from the seriousness of the subject; instead, it makes the cruelty and self-interest beneath the political performance stand out more sharply.

Near the end, Fernando echoes Macbeth with the line, “There is judgment here.” It captures the question running through the novel: can anyone participate in corruption and remain untouched by it? Beneath this satire is a dark story about power and the danger of believing that someone else will always pay the price.

I came away feeling that Outpost of Empire is a valuable and thought-provoking book, particularly in today’s political and social climate. It has a distinctive voice, and the questions it raises about power, responsibility and foreign interference remain relevant far beyond the fictional borders of Lackadasia.
About the Author

Shavindra Fernando has worked as a reporter. He read English & Classics at the University of Peradeniya in Sri Lanka, and took International Relations & Ethnic Politics as a Reuter Fellow at the University of Oxford in England. The Vague Poetess (1999)was his first novel.

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