Sophie Marques and Tristan Bernard have brought the idea of applying the art of war by Sun Tzu in understanding the concepts of mathematics. Beyond that, they have categorically elaborated how the water philosophy and Observation approaches could be fundamental in learning and understanding deeply the concepts of Mathematics.
Sun Tzu said: “The good fighters of old first put themselves beyond defeat, and then waited for an opportunity of defeating the enemy”, The Art of War.
To be beyond defeat as Sun Tzu stated is not to be secure in concepts that are well grasped, but to be flexible and open to the world. It is therefore such that not everything is known to any one individual but rather every individual has the ability to teach and be taught new knowledge. Take the example of a mathematics professor who is an expert in his field but views his perspective as the only correct view point. He has placed himself securely in that which he understands by disregarding alternative perspectives creating a void where understanding does not exist. His one-sided knowledge thus feeds his ignorance. It then follows that with increased knowledge in a field comes confidence. This confidence makes you sure of your own viewpoint but also robs you of your once-held caution. It is so that you might intellectually isolate yourself. It is rather more appropriate to be metaphorically like water, calm, powerful, and formless.
Become like water
From a warrior standpoint, being like water means being able to adapt to whichever situation you find yourself in. In mathematics, you can interpret this adaptability as having beforehand insight into the work content and thus having a plan of attack to succeed. In pursuit of adaptability, you may develop yourself to deal with the unpredictable by taking conscious observations of the material or circumstance before you, therefore deepening your insight into these challenges. You must however also become observant of your own behavior to see correlations between circumstances and how your own actions influence them. Moreover, water takes the path of least resistance and as such each situation or circumstance may in fact have a natural or effortless solution based on an alternative approach. Try to see each situation from many angles as well as your role in them before you move. In mathematics, this could mean that the student should not rush to the answer but consider the questions from many different perspectives and find an easier approach to solve the problem. It therefore follows that answering the problem is not the essence of the exercise but the process to arrive at a correct conclusion. This too applies to the metaphorical idea of water as it is in constant motion: evaporating, flowing, condensing and so this reflects the importance of the journey, the change you take on.
In addition, water has a reflective surface which shows a mirrored image of the world around it and thus reflects the dual nature of knowledge and understanding. There is surface knowledge which is the reflection and there is deep knowledge, that which is hidden in the depths. It is also true then, that there is incomplete and complete understanding of concepts and ideas. So, to be like water is to reflect back with a mirrored surface that which you observe. In essence, become a great observer, and study your opponent. Your examination of them will unveil the shadow of mystery in which they dwell while providing the key to their defeat. Water hides its secrets and often we cannot see what is in the depths only a reflection of the world above its surface. It also distorts its contents through refraction and flows to fill its environment. From a mathematical perceptive, this would mean reflecting back the concept presented to you and this can only be done with understanding. On the other hand, you need to treat what is given to you in mathematics as only the surface of the concept, the mirror. In order to understand the idea and use it properly you will need to go deeper and question what is given to you, you will need to search the depths. That is where you will find what you don’t understand and subsequently need to be aware of.
When it is said to be like water, it means to take on the attributes of water. Water is immensely powerful, but we often do not believe so. One cubic meter of water weighs one metric ton, and flowing water is one of the most powerful forces in nature. Water always finds the easiest path forward, it flows downhill, around the densest of stone, and has enough force to destroy what is in its path. The power of water acts sometimes slowly, for instance it may polish rough stones and form pebbles given enough time. In addition, with depth water becomes powerful, so with more depth in understanding what you don’t understand you can become successful in mathematics and in other areas. Your success in mathematics depends on your consistency to make progress. Water does not appear as a powerful force when still, but in motion its power is beyond question and so it too is true in your case if you continue little by little to your goal. You however much like water need to voluntarily accept the challenge and you need to understand that success like understanding is a process. You therefore will misunderstand and miss key concepts and this will be challenging to overcome, but like water you need to trust that you are able to succeed and slowly understand given time and small steps forward. Therefore, understand the concept that you don’t understand everything, and that it is ok. It is a gateway into learning many new things.
Understanding Within Observation
Observation is most potent when what we are observing is being seen for the first time but is robbed of that potency with repeated exposure to the same stimulus. In essence, once we become comfortable, we become complaisant and allow the conscious mind to disengage with the tasks being performed. For example, take driving a car; have you ever driven to a place and then wondered how you arrived at your destination? You essentially have allowed muscle memory to complete the task of driving without engaging the conscious mind. There are benefits to this approach, for instance, it allows us to complete simple tasks efficiently while leaving the mind available for more intricate problem-solving. Then there are drawbacks on the other hand which are simple errors accumulating in such tasks due to the conscious mind not overseeing said tasks’ completion. This then leads into the question of presence:
How present are we in our own lives?
It most certainly can be viewed as a significant question, as it asks to what extent we are passengers to muscle memory vs active participants in our lives. The answer to such a question can only come with the active observation of oneself, to become aware of that which is unconscious.
Observation Within Mathematics
It can be agreed that observation is a skill that needs practice to fully develop into its most potent form. Therefore perhaps, mathematics is the perfect incubator and test bed for such a skill to be developed.
Mathematics is structured in such a way that new problems become old problems while simultaneously being replaced in said ranking of old and new. This implies that stimulus continually changes ranking from new (heightened observation) to ordinary (simply solved). This progression thus invites complacency in problem-solving. For instance, take the example of someone who is viewed as intelligent in mathematics but gets all the simple calculations wrong but the complex problems correct. This disparity in correct answers vs incorrect answers and complex vs non-complex might imply the shift between muscle memory and conscious observation within the task of problem solving.
To further elaborate, they might have become complacent within simple problems due to their “simple” moniker and thus completed the task of calculation without the involvement of the conscious mind. In addition, all mathematics problems consist of some form or other of simple calculations and a wrong answer is still a wrong answer regardless if you know how to solve it or not. It thus creates a perfect feedback loop to assess your self-observation. An increase in simple errors would imply a decrease in conscious engagement and vice versa.
Nevertheless, there are two additional situations in mathematics. Firstly, you could have a given understanding of a concept at some point but with repetition and using only muscle memory will discover that after sometime you have lost that deep knowledge required to truly understand the problem. In the second situation, you might have built confidence in mathematics by relying solely on muscle memory to mimicking proficiency through procedural memorization of problem solving. The danger in this is that you may not be aware that there is more to be understood and so when you arrive at a higher grade you are confused and subsequently frustrated because you thought you were good at mathematics but you were only good at mimicking the steps presented to you.
Finally, with practicing self-observation it might become clear what the solution is to unsolvable problems. For instance, take the example of not understanding the topic in mathematics; the question is usually why? But there might not be a good answer until you observe yourself. In self-observation what is unclear becomes clear and as such perhaps the reason you do not understand something in class is because of financial stress, late nights, a bad lecturer or simply not paying attention. Now that you are present in that situation, you are made aware of the issues that exist unseen and can solve the problems to the best of your ability and means to achieve the greatest success.
Mathematics Within Understanding
Therefore, to understand that which you do not understand requires the engagement of the powers of conscious observation. To use the tools that exist within the structures of mathematics to facilitate the growth and development of skills which are useful beyond the scope of mathematics alone.
Treat all things as unknowns, enigmas and problems which must be solved so that you might observe them with fresh eyes and never become complacent. Observe the world around you like an inquisitive child exploring so that there is no unknown within what is known.
Tristan Barnard
BSc Wood & Wood Product Science student, Stellenbosch University, South Africa
DR. SOPHIE MARQUES
Department of Mathematical Sciences, Stellenbosch University, South Africa
