In this article, Tristan Barnard, Sophie Marques and Rivan Rughubar delve into the paradox of recognizing the unknown. They explore how understanding the limits of our knowledge is crucial for intellectual growth. Employing techniques from mathematics, the authors illustrate how to identify and address gaps in comprehension. By doing so, they show how we can bridge the divide between the known and unknown, enhancing our understanding and taking actionable steps toward continuous learning and discovery.
At any given moment, our understanding is constrained by the limits of our knowledge. No matter how much we comprehend a subject, there will always be aspects that remain beyond our grasp. This raises the question:
How can we become aware of what we don’t understand? Is it not paradoxical to attempt to comprehend the unknown?
In essence, the challenge lies in bringing the realm of the unknown into our awareness. Fortunately, there are techniques, frequently employed in the field of mathematics, mainly based on the development of logic, that can effectively enhance our awareness of what we don’t understand, empowering us to take meaningful action.
By employing these techniques, we can bridge the gap between what we currently comprehend and the unexplored territories of knowledge. This enables us to identify areas of deficiency and actively address them. Instead of being hindered by the paradoxical nature of the unknown, we can use these tools to expand our understanding and make tangible progress in our quest for knowledge.
Exploring the Depths of Understanding: Unraveling the Complexity.
Is understanding a subjective experience that varies from person to person? Our journey through different levels of understanding is incredibly intricate. At any given moment, we might feel confident in our comprehension of a particular subject, but acquiring new insights can sometimes challenge and even question that understanding, giving the impression that our level of understanding has diminished. However, in reality, every additional piece of understanding contributes to our overall growth in knowledge. But how can we begin to grasp what we don’t understand? This is where mathematics proves to be an invaluable tool.
Let’s consider two examples, one where our understanding of the physical world can help us find flaws in our conceptual understanding and one where our conceptual understanding can help us learn more about the physical world.
Example 1: Imagine you are tasked with calculating the length of a ladder leaning against a wall using trigonometry. You could likely take a guess at the length of the ladder just by looking at it, but trigonometry allows you to find a more accurate answer. You perform the calculation, and your answer turns out to be a negative number. You know this can’t be true as the ladder is in front of you and so you must conclude that there is an error in your calculations. At this point, you can delve deeper into the subject, striving to uncover the root of what you don’t understand or identify the missing piece. By doing so, you can further enhance your understanding and bridge the gaps in your knowledge.
Example 2: Early humans believed that the Earth was flat. This was an idea that went unchallenged for most people as it correlated with their lived experience. However, from as far back as the ancient Greeks, around 500 BC, we have claims and experiments proving that the surface of the Earth is curved. They were able to use mathematical and geometrical observations to change their understanding of the physical world and it is a practice we still follow today. In this case they believed their understanding of the mathematical rules to be more concrete than that of the physical world around them and by allowing their assumptions to be challenged they made a great discovery.
If we fail to understand what we don’t understand, we may find ourselves in a situation where we believe we understand everything because it doesn’t even occur to us that something is missing. This can be observed in a math class where students confidently claim to have performed well on a test, only to be faced with disappointing results, leading to deep frustration. Not understanding what we don’t understand can be an isolating and vexing state to be in.
We can explore a linked concept: understanding what we understand. As Professor Volmink highlighted when presented with this concept, we can envision this as a two-dimensional space. One axis represents knowledge, ranging from non-understanding to understanding (assuming there is some form of understanding that is not relative to anyone), while the other axis represents our perception of our own understanding at a given time, also ranging from non-understanding to understanding.

Understanding what we understand may initially seem paradoxical. How could we not comprehend what we already understand? One possibility is that our understanding might reside at an intuitive level. We know what to do and when to do it, but we may lack a deeper understanding of why we do it that way—it simply feels right, perhaps due to our past experiences. In this case, we need to consciously deepen our understanding, becoming aware of the underlying principles.
Now, understanding what we understand serves the purpose of formalizing our understanding in our conscious mind. This can be exemplified by our ability to teach others a particular concept. When others grasp our explanations, it increases the likelihood that we truly understand what we are conveying. This is why teaching others is considered the pinnacle of the teaching and learning process. The complexity lies in the fact that while our understanding can continue to grow, we cannot fully control its boundaries. However, mathematics can be an ideal place to learn how to embrace the unknown positively and acquire the tools to unravel what we understand from what we don’t.
The Multifaceted Nature of Mathematical Understanding: Individual Perspectives and Intersections
Mathematics, as a subject, encompasses a vast array of concepts and techniques, and it is intriguing to ponder whether each person’s understanding of mathematics is uniform. The level of understanding can greatly vary between individuals, leading to potential contradictions when perspectives diverge or fail to intersect. This intersection of understanding becomes crucial in reconciling different viewpoints and embracing insights gained from others.
One compelling aspect to consider is how different methods of approaching mathematics can shape our perception and understanding of the subject. Each individual may employ unique strategies, problem-solving techniques, and conceptual frameworks, resulting in distinct perspectives. These varying perspectives contribute to the richness and diversity within the mathematical landscape, providing different lenses through which to explore and appreciate mathematical concepts.
Furthermore, the process of confronting alternative perspectives and engaging in intellectual discourse is akin to cryptography—a means of deciphering the hidden treasures within mathematics. When individuals with different levels of understanding come together, engaging in discussions and debates, they shed light on new facets of mathematical thinking. This intellectual confrontation acts as a catalyst for personal growth, pushing the boundaries of individual comprehension and fostering a deeper understanding of the subject.
In this dynamic environment, the interplay between diverse perspectives and the challenges presented by confronting others’ viewpoints serve as the crucible for intellectual growth and development. It is within these crucibles that the “hard light” of mathematics illuminates new insights and expands our collective understanding of this intricate discipline.
Ultimately, embracing the diverse spectrum of mathematical understanding, acknowledging intersecting levels of comprehension, and engaging in constructive confrontations allow us to embark on a collective journey of discovery, pushing the boundaries of knowledge and opening new avenues of exploration in the captivating realm of mathematics.
The Multifaceted Nature of Mathematical Understanding: Perspectives, Confrontations, and Collective Growth
Mathematical understanding is a diverse landscape, akin to strata in the fabric of time. Each individual possesses a unique understanding, shaped by their experiences and influenced by their personal logic. These understandings often intersect in complex ways, forming an intricate web of comprehension. However, it is crucial to recognize that the nature of understanding is non-comparable, as there are always aspects that elude our grasp, residing outside our reach. By embracing this complexity and adopting a flexible mindset, we can shift our perspectives, learn to see from various angles, and enhance our understanding. Mathematics, with its myriad of methods and approaches, plays a pivotal role in fostering this flexible mindset.
Fostering Flexibility of Mind:
In traditional education, the emphasis often lies on singular methods and algorithmic thinking. However, it is essential to cultivate a more versatile approach, one that encourages multiple perspectives. Mathematics, as a discipline, offers a rich array of methods for problem-solving, encouraging students to explore diverse avenues. By deviating from a single rigid path and engaging with different strategies, students develop a more flexible mind, capable of adapting to various perspectives and tackling complex problems with greater creativity.
The Power of Confrontation:
In the realm of mathematics, individual progress is not solely achieved in isolation. Rather, it emerges through constructive confrontations with other minds. While mathematics is often portrayed as an individual endeavor, the essence of its progress lies in the collective engagement of diverse perspectives. By encountering minds that perceive the world differently, we gain fresh insights and broaden our understanding. These new perspectives do not negate our existing understanding but build upon it, expanding our collective knowledge and deepening our comprehension of complex mathematical concepts.
Levels of Understanding:
Understanding operates on multiple levels, ranging from the individual to the collective and encompassing various educational backgrounds. The intricacy and beauty lie in the fact that each person possesses a unique understanding that cannot be fully perceived or appreciated by others. This inherent uniqueness adds depth to the tapestry of human comprehension, creating a mosaic of diverse perspectives. It is this multiplicity of understanding that enriches the collective discourse and propels the advancement of mathematics.
Conclusion:
Imagine a world where individuals embrace diverse perspectives with an unwavering commitment to understanding, free from judgment. In this ideal society, people actively seek to expand their horizons by empathetically exploring alternative viewpoints, fostering an environment of open-mindedness and intellectual growth. In the pursuit of knowledge, we must remain cognizant of the potential fallibility of long-held beliefs. The hidden truths that evade our understanding offer exciting opportunities for growth and exploration. By embracing uncertainty and actively questioning assumptions, we embark on a remarkable journey of discovery, unraveling the mysteries of the universe and expanding the frontiers of human knowledge. In this intricate landscape of mathematical understanding, we are reminded of the beauty inherent in our individual perceptions. Each of us holds a fragment of understanding that is exclusive to our own experiences and perspectives. The elusive nature of this understanding fosters curiosity and drives us to explore further. By embracing diverse perspectives, engaging in constructive confrontations, and appreciating the collective nature of mathematical progress, we embark on a journey of profound discovery and growth. Mathematics, with its vast toolbox of methods, acts as a catalyst, fostering flexibility of mind and unveiling new dimensions of understanding.

Tristan Barnard
BSc Wood & Wood Product Science student, Stellenbosch University, South Africa

DR. SOPHIE MARQUES
Department of Mathematical Sciences, Stellenbosch University, South Africa

Rivan Rughubar.