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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

Buddhist insights into the extended mind thesis – Some observations

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It is both an honour and a pleasure to address you on this occasion as we gather to celebrate International Philosophy Day. Established by UNESCO and supported by the United Nations, this day serves as a global reminder that philosophy is not merely an academic discipline confined to universities or scholarly journals. It is, rather, a critical human practice—one that enables societies to reflect upon themselves, to question inherited assumptions, and to navigate periods of intellectual, technological, and moral transformation.

In moments of rapid change, philosophy performs a particularly vital role. It slows us down. It invites us to ask not only how things work, but what they mean, why they matter, and how we ought to live. I therefore wish to begin by expressing my appreciation to UNESCO, the United Nations, and the organisers of this year’s programme for sustaining this tradition and for selecting a theme that invites sustained reflection on mind, consciousness, and human agency.

We inhabit a world increasingly shaped by artificial intelligence, neuroscience, cognitive science, and digital technologies. These developments are not neutral. They reshape how we think, how we communicate, how we remember, and even how we imagine ourselves. As machines simulate cognitive functions once thought uniquely human, we are compelled to ask foundational philosophical questions anew:

What is the mind? Where does thinking occur? Is cognition something enclosed within the brain, or does it arise through our bodily engagement with the world? And what does it mean to be an ethical and responsible agent in a technologically extended environment?

Sri Lanka’s Philosophical Inheritance

On a day such as this, it is especially appropriate to recall that Sri Lanka possesses a long and distinguished tradition of philosophical reflection. From early Buddhist scholasticism to modern comparative philosophy, Sri Lankan thinkers have consistently engaged questions concerning knowledge, consciousness, suffering, agency, and liberation.

Within this modern intellectual history, the University of Peradeniya occupies a unique place. It has served as a centre where Buddhist philosophy, Western thought, psychology, and logic have met in creative dialogue. Scholars such as T. R. V. Murti, K. N. Jayatilleke, Padmasiri de Silva, R. D. Gunaratne, and Sarathchandra did not merely interpret Buddhist texts; they brought them into conversation with global philosophy, thereby enriching both traditions.

It is within this intellectual lineage—and with deep respect for it—that I offer the reflections that follow.

Setting the Philosophical Problem

My topic today is “Embodied Cognition and Viññāṇasota: Buddhist Insights on the Extended Mind Thesis – Some Observations.” This is not a purely historical inquiry. It is an attempt to bring Buddhist philosophy into dialogue with some of the most pressing debates in contemporary philosophy of mind and cognitive science.

At the centre of these debates lies a deceptively simple question: Where is the mind?

For much of modern philosophy, the dominant answer was clear: the mind resides inside the head. Thinking was understood as an internal process, private and hidden, occurring within the boundaries of the skull. The body was often treated as a mere vessel, and the world as an external stage upon which cognition operated.

However, this picture has increasingly come under pressure.

The Extended Mind Thesis and the 4E Turn

One of the most influential challenges to this internalist model is the Extended Mind Thesis, proposed by Andy Clark and David Chalmers. Their argument is provocative but deceptively simple: if an external tool performs the same functional role as a cognitive process inside the brain, then it should be considered part of the mind itself.

From this insight emerges the now well-known 4E framework, according to which cognition is:

Embodied – shaped by the structure and capacities of the body

Embedded – situated within physical, social, and cultural environments

Enactive – constituted through action and interaction

Extended – distributed across tools, artefacts, and practices

This framework invites us to rethink the mind not as a thing, but as an activity—something we do, rather than something we have.

Earlier Western Challenges to Internalism

It is important to note that this critique of the “mind in the head” model did not begin with cognitive science. It has deep philosophical roots.

Ludwig Wittgenstein

famously warned philosophers against imagining thought as something occurring in a hidden inner space. Such metaphors, he suggested, mystify rather than clarify our understanding of mind.

Similarly, Franz Brentano’s notion of intentionality—his claim that all mental states are about something—shifted attention away from inner substances toward relational processes. This insight shaped Husserl’s phenomenology, where consciousness is always world-directed, and Freud’s psychoanalysis, where mental life is dynamic, conflicted, and socially embedded.

Together, these thinkers prepared the conceptual ground for a more process-oriented, relational understanding of mind.

Varela and the Enactive Turn

A decisive moment in this shift came with Francisco J. Varela, whose work on enactivism challenged computational models of mind. For Varela, cognition is not the passive representation of a pre-given world, but the active bringing forth of meaning through embodied engagement.

Cognition, on this view, arises from the dynamic coupling of organism and environment. Importantly, Varela explicitly acknowledged his intellectual debt to Buddhist philosophy, particularly its insights into impermanence, non-self, and dependent origination.

Buddhist Philosophy and the Minding Process

Buddhist thought offers a remarkably sophisticated account of mind—one that is non-substantialist, relational, and processual. Across its diverse traditions, we find a consistent emphasis on mind as dependently arisen, embodied through the six sense bases, and shaped by intention and contact.

Crucially, Buddhism does not speak of a static “mind-entity”. Instead, it employs metaphors of streams, flows, and continuities, suggesting a dynamic process unfolding in relation to conditions.

Key Buddhist Concepts for Contemporary Dialogue

Let me now highlight several Buddhist concepts that are particularly relevant to contemporary discussions of embodied and extended cognition.

The notion of prapañca, as elaborated by Bhikkhu Ñāṇananda, captures the mind’s tendency toward conceptual proliferation. Through naming, interpretation, and narrative construction, the mind extends itself, creating entire experiential worlds. This is not merely a linguistic process; it is an existential one.

The Abhidhamma concept of viññāṇasota, the stream of consciousness, rejects the idea of an inner mental core. Consciousness arises and ceases moment by moment, dependent on conditions—much like a river that has no fixed identity apart from its flow.

The Yogācāra doctrine of ālayaviññāṇa adds a further dimension, recognising deep-seated dispositions, habits, and affective tendencies accumulated through experience. This anticipates modern discussions of implicit cognition, embodied memory, and learned behaviour.

Finally, the Buddhist distinction between mindful and unmindful cognition reveals a layered model of mental life—one that resonates strongly with contemporary dual-process theories.

A Buddhist Cognitive Ecology

Taken together, these insights point toward a Buddhist cognitive ecology in which mind is not an inner object but a relational activity unfolding across body, world, history, and practice.

As the Buddha famously observed, “In this fathom-long body, with its perceptions and thoughts, I declare there is the world.” This is perhaps one of the earliest and most profound articulations of an embodied, enacted, and extended conception of mind.

Conclusion

The Extended Mind Thesis challenges the idea that the mind is confined within the skull. Buddhist philosophy goes further. It invites us to reconsider whether the mind was ever “inside” to begin with.

In an age shaped by artificial intelligence, cognitive technologies, and digital environments, this question is not merely theoretical. It is ethically urgent. How we understand mind shapes how we design technologies, structure societies, and conceive human responsibility.

Buddhist philosophy offers not only conceptual clarity but also ethical guidance—reminding us that cognition is inseparable from suffering, intention, and liberation.

Dr. Charitha Herath is a former Member of Parliament of Sri Lanka (2020–2024) and an academic philosopher. Prior to entering Parliament, he served as Professor (Chair) of Philosophy at the University of Peradeniya. He was Chairman of the Committee on Public Enterprises (COPE) from 2020 to 2022, playing a key role in parliamentary oversight of public finance and state institutions. Dr. Herath previously served as Secretary to the Ministry of Mass Media and Information (2013–2015) and is the Founder and Chair of Nexus Research Group, a platform for interdisciplinary research, policy dialogue, and public intellectual engagement.

He holds a BA from the University of Peradeniya (Sri Lanka), MA degrees from Sichuan University (China) and Ohio University (USA), and a PhD from the University of Kelaniya (Sri Lanka).

(This article has been adapted from the keynote address delivered
by Dr. Charitha Herath
at the International Philosophy Day Conference at the University of Peradeniya.)

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Opinion

We do not want to be press-ganged 

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Reference ,the Indian High Commissioner’s recent comments ( The Island, 9th Jan. ) on strong India-Sri Lanka relationship and the assistance granted on recovering from the financial collapse of Sri Lanka and yet again for cyclone recovery., Sri Lankans should express their  thanks to India for standing up as a friendly neighbour.

On the Defence Cooperation agreement, the Indian High Commissioner’s assertion was that there was nothing beyond that which had been included in the text. But, dear High Commissioner, we Sri Lankans have burnt our fingers when we signed agreements with the European nations who invaded our country; they took our leaders around the Mulberry bush and made our nation pay a very high price by controlling our destiny for hundreds of years. When the Opposition parties in the Parliament requested the Sri Lankan government to reveal the contents of the Defence agreements signed with India as per the prevalent common practice, the government’s strange response was  that India did not want them disclosed.

Even the terms of the one-sided infamous Indo-Sri Lanka agreement, signed in 1987, were disclosed to the public.

Mr. High Commissioner, we are not satisfied with your reply as we are weak, economically, and unable to clearly understand your “India’s Neighbourhood First and  Mahasagar policies” . We need the details of the defence agreements signed with our government, early.

 

RANJITH SOYSA 

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Opinion

When will we learn?

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At every election—general or presidential—we do not truly vote, we simply outvote. We push out the incumbent and bring in another, whether recycled from the past or presented as “fresh.” The last time, we chose a newcomer who had spent years criticising others, conveniently ignoring the centuries of damage they inflicted during successive governments. Only now do we realise that governing is far more difficult than criticising.

There is a saying: “Even with elephants, you cannot bring back the wisdom that has passed.” But are we learning? Among our legislators, there have been individuals accused of murder, fraud, and countless illegal acts. True, the courts did not punish them—but are we so blind as to remain naive in the face of such allegations? These fraudsters and criminals, and any sane citizen living in this decade, cannot deny those realities.

Meanwhile, many of our compatriots abroad, living comfortably with their families, ignore these past crimes with blind devotion and campaign for different parties. For most of us, the wish during an election is not the welfare of the country, but simply to send our personal favourite to the council. The clearest example was the election of a teledrama actress—someone who did not even understand the Constitution—over experienced and honest politicians.

It is time to stop this bogus hero worship. Vote not for personalities, but for the country. Vote for integrity, for competence, and for the future we deserve.

 

Deshapriya Rajapaksha

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