On mathematical thinking and its applications

"Mathematics is the music of reason."

James Joseph Sylvester

Mathematics – that school subject that sometimes causes a cold shiver to run down the spine of many a student. Contrary to what many people think, math is not a very difficult subject and one does not need to have an aptitude for numbers in order to excel at it. In this essay, I will like to throw some light on the relevance of math in our school curriculum, discuss its numerous applications, and suggest a method in which the teaching of mathematics should be administered in our schools and colleges.

Math is the science of quantity and number. It is omnipresent in everyday life. Take a good look at the anatomy of most living beings and you will see symmetry; life cycles of animals and plants, the evolution of natural processes like weather patterns and seasonal cycles, and the motion of heavenly bodies (like the planets around the sun) bear testimony to the importance of periodicity in nature. Differential equations are employed in the modeling of increase and plummet in stock prices, prime numbers are used to enforce security in our computer systems, and optimization of industrial processes are achieved with the use of calculus. From an early age, mathematical thinking becomes so deeply ingrained in the very fabric of our thought processes, so much so that we become oblivious to how much we depend on it. For example, before we even start going to pre-school, we are able to recognize differences in quantity, sort objects, and identify patterns (like the interchange between day and night). I cannot overemphasize the key role math plays in our society and why it should be regarded as a sublime tool we need to master in order to better understand how our world operates.

Mathematical thinking permeates many areas of modern society. In Law, logical thinking aids lawyers to formulate factual and rational arguments that can convince members of the jury to acquit their clients. This is because a large part of mathematics teaches individuals how to construct flawless proofs. The language of math is extremely precise. Over the years, mathematicians have developed a consistent system of logic and terminology that has helped to resolve ambiguity in mathematical parlance. This approach has been incorporated into several fields like computing, linguistics, engineering just to name a few. For example, in computing, computers would never have been built if the programming languages which are used to operate them were not exact. Due to the sometimes imprecise nature of natural language, linguists use a technique called part-of-speech tagging to disambiguate between similar meanings of a sentence. Beyond precision and rigor, mathematical thinking helps us to analyze and solve problems for which we do not have a standard procedure. This aspect helps explain why major employers of nowadays value workers who have strong analytical thinking skills – those who are able to “think outside the box” when faced with novel problems. This ability helped Stanford graduate students — Larry Page and Sergey Brin — to develop a new, efficient mathematical procedure (called the PageRank algorithm) for web searches, which led to their creation of Google in 1998.

Do you remember the approach you took when you were told find the roots of a cubic polynomial? What you likely did was to guess one of its roots, and then use it to obtain a quadratic polynomial which you could easily factorize. Well, you weren’t aware that what you were using recursive thinking to solve your problem. Recursive thinking, commonly known as the divide-and-conquer strategy, involves breaking a large problem into smaller chunks (mini-copies of the original) which are easier to solve. Recursive thinking is a derivative of recursion – the latter refers to the repeated application of a procedure on itself. The use of recursion is pervasive in our world. The most efficient sorting and search algorithms are recursive in nature. Sorting algorithms help keep the alphabetical ordering of names in our phone contact list, priority queues control interrupts so that phone users are able to pick up calls even when using apps, compression algorithms are used to reduce the size of memory used to store files, thanks to the power of recursion. Fractals, which are objects composed of repeated patterns of some unit structure, are constructed using recursion. Capillary branches of our blood vessels and dendrites of neurons, snowflakes, tree branches, lightning bolts, and the awe-inspiring shell of the Nautilus snail, are all examples of fractals that can be found in nature. Our blood capillaries take advantage of their fractal (or branched network nature) to carry out the efficient distribution of blood to far-off tissues. Recursion has a place in games too. In chess, for example, the eight-queens problem can be solved using recursive thinking. Indeed, recursion is pervasive, and we must learn to accommodate it in our problem-solving toolkits.

Is it a worthwhile use of time to elaborate on how we can amend the teaching of math? Given the above arguments, Yes! How then should mathematics be taught? My proposal is that teachers should change the way the subject is presented to students. During the teaching of mathematics, more emphasis should be placed on thinking through a problem, instead of applications of procedures or formulae. As the old adage goes: “Give a man a fish to eat, and he’ll eat for a day. Teach a man to fish and he’ll eat for a lifetime.” When students are taught to simply apply formulae, they lose their creativity, and as such, they tend to shy away from solving problems that are different from those they’ve encountered before. Mathematics is the science of patterns. The multitude of problems in our world can be categorized into different classes; problems belonging to a particular class require a certain pattern of thought to arrive at their solutions. These patterns of thought represent the various branches of mathematics like logic, probability, etc. Students should, therefore, have a solid understanding of the different classes of problems and the thinking patterns associated with each class, so that they can go beyond the application of formulae to the development of new, sophisticated techniques to solve novel problems they meet in future. In our world of bewildering complexity, where the breadth or scope of problems is seemingly endless, only those who are able to learn, unlearn and relearn new ways of approaching problems will soar above the challenges the future holds. And I will argue that a fair amount of maturity in mathematical thinking is required to develop the aforementioned abilities.

I believe its time we moved on from whining about how challenging math is to appreciating its efficacy so that we can harness the power of mathematical thinking to find the best solutions to our diverse local and global impediments. To wrap up this essay, I leave you with the quote below:

"Art begins in imitation and ends in innovation."

Give us the chance to dream big again!

“Education is life itself rather than mere preparation for life.”

J.W. Norman

“Give us the chance to dream again!” This is a clarion call I make to all current African governments and even more so to that of my country Cameroon. When we all were kids, we harbored and cherished the thoughts of being great men and women in the future, who will leave a mark on our generation. Little did we know, that as young Africans, the path to our beloved dreams would be wearisome and full of obstacles of all sorts. I think some of these hurdles that stand in our way are common to all societies ( like lack of finances to pay for education ). But sad to say, the toughness and severity of these challenges are particularly higher in African countries. For example, though there are numerous academic establishments in most African societies, a good number of those who complete their education are not leveraging their full potential, and worse still, unemployment has become commonplace. More often than not, we can trace the cause of this issue back to our incompetent education systems.

We need to revamp our education systems. Though the age of rote learning is now history in developed countries, it is presently a stark reality in most African countries. Albert Einstein once said: “A society’s competitive advantage will not come from how well its schools teach the periodic and multiplication tables, but from how well they stimulate imagination and creativity.” Most of the technological inventions we enjoy nowadays were inspired by individuals who tried to satisfy their curiosity, whose desires to push the current boundaries of knowledge were rewarded with fresh insight on the unknown workings of our world. We need to put in place an education system that will train people how to think critically, one that will provide a conducive atmosphere for individuals to question the status quo, as well as promote problem-solving skills — which are all the more necessary in today’s world of overwhelming complexity. Rote learning isn’t entirely bad; it helps us get a good mastery of foundational knowledge. This fact is enunciated by the age-old proverb: “Repetition is the mother of all learning”. What we should guard against is the act of making mindless memorization the alpha and omega of the training and evaluation of individuals in our academic institutions, as it will lead us to nowhere in this day and age.

I have experienced first-hand what studying in a poorly developed university system can be like. I spent two years in an engineering school in my home country Cameroon. During those years, I used up almost all of my time memorizing equations and theorems, instead of getting hands-on practical training — that usually makes one savor the excitement that comes with being in such an institution. I often wondered if I would deserve to be called an engineer at the end of my studies, given that the technical aspects of my training were downgraded as opposed to the conceptual ones. By the end of my two-year study period, I had nothing to show for other than the ability to memorize lots of information at a go. From the look of things, I came to realize the enormous potency poor educational systems have to slow down the progress of our African economies. As was the case with my aforementioned engineering studies, half-baked training of individuals usually results in graduates who lack a sense of innovation and dynamism, thus giving rise to an amateurish workforce and a stagnant economy. According to a recent United Nations report, it is estimated that by 2050 there will be about 2 billion people living on the African continent. African countries are therefore in dire need of restructuring their systems of general and vocational training, so as to produce graduates that are capable of using their expertise and problem-solving abilities to address the financial and economic needs of the ever-growing population.

Thankfully, I did not complete my studies in the above-stated engineering school. I moved to another university, this time in the UK, to study computer science. So far, I have found my studies very comfortable (though seldom challenging), I love my professors and I am amazed at the sheer number of student support facilities available at my current university. Everything I now learn is geared towards problem-solving and real-world applications. Although I am pleased with how well my education is going, I sometimes cannot withhold my thoughts from wondering about the state of affairs back home. Occasionally, though, I find relief in knowing that some countries like Ghana and Rwanda are making impressive strides in the remodeling of their education systems and that a good number of African youths are embracing entrepreneurial thinking. To end my discourse, I pose the following questions to our African leaders: Must we emigrate from our home countries in order to receive sound academic instruction? Shouldn’t they (our leaders) give reason a chance once and for all? Is it not time they woke up from slumber? For how long should we keep on waiting for reforms? Shouldn’t the innocent African children of today be given the chance and space to nurture their God-given talents in order to fulfill their dreams and contribute to the improvement of the livelihood of our existing societies? The answers are blowing in the wind.

A tour through the field of Computer Science

Computers don’t introduce order anywhere as much as they expose opportunities.

Alan Perlis, ‘Epigrams in programming’. 

According to the layman, computer science is the study of the principles and use of computers. Also, it is not uncommon to hear many people define computer science as the study of algorithms and various programming paradigms, a subject whose sole aim is to provide the machinery needed to build computer software applications — which is but a  fraction of what computer scientists do. Computer science has rapidly evolved from the building of electromechanical and later digital electronic computer systems to a broader discipline called informatics which studies how information is processed in natural and artificial (or engineered) systems. The numerous applications of informatics have become more and more pervasive in modern day technology such that it is a mind-boggling task for us to imagine how a world without computer systems might be. From system software such as operating system kernels to large applications like Microsoft Excel, LANS to the globe-spanning internet and the Web, embedded systems to supercomputers and artificial neural networks, computer systems have transformed the faces of the 20th and 21st centuries. With this in mind, it will be helpful for us to understand how we got to where we are by looking at how the different branches of informatics have shaped the technology landscape.

A statue of Alan Turing at Bletchley Park

Alan Turing, a British mathematician (considered by many as the father of computer science) and the American logician Alonzo Church, together laid the theoretical foundations for computer science in the early 1930s. Turing came up with the concepts of memory, stored program computers and Turing machines, which several individuals like the polymath Von Neumann further developed. Not long after that, some countries around the world like the US embarked on various projects to build Turing-complete electromechanical computing systems that could facilitate long, arduous numerical calculations in different fields like astronomy, physics, and accounting just to name a few. Great strides were made in these projects due to the invention of the transistor and in later years, integrated circuits.

Let me break away from this point for a moment to mention that I regard the invention of digital computers as a remarkable feat of electrical engineering. Millions of transistors are employed in the production of microprocessors and the motherboard as a whole and built upon several layers of hardware abstraction and system software, are the software applications — written by ingenious individuals — like web browsers which we are able to use for our daily tasks. We can trace the emergence of modern day computers back to Colossus and ENIAC, which were among the first digital programmable stored program computers (so many adjectives 🙂 ). The development of several easy-to-use programming languages and software engineering methodologies spurred the proliferation of software businesses like Microsoft, Apple, and Oracle. So far, most if not all of the success of computer systems could be attributed to efficient computing algorithms, programming language theory, and electronic engineering.

Google Home, a virtual assistant

 

Enter AI. The term “artificial intelligence” was coined in 1955 by the American computer scientist John McCarthy who invented time-sharing (that led to the internet and cloud computing) and the Lisp programming language. AI, though a branch of informatics, draws heavily upon material from the fields of linguistics, psychology and cognitive science. It seeks to understand the processes by which information is received, interpreted, analyzed and acted upon in natural systems like the human brain in order to mimic these processes in engineered systems. Needless to say, AI has made contemporary computer systems far smarter than their predecessors as they are able to carry out natural language translation (e.g. Google Translate), speech processing through virtual assistants (Alexa, Siri, Cortana, Google Home), visualize their environments (robots are a typical example) and play games such as chess and Go. In fact, so successful has AI been that DeepMind’s AlphaGo software though fed only with the rules of the Go game was able through self-learning techniques to beat a human professional Go champion Lee Sedol in 2016. Machine learning — be it supervised, unsupervised or reinforced, agent-based systems, neural networks, algorithmic game theory, and data mining are key areas of AI that are promising to revolutionize the technologies of the future.

AlphaGo versus Lee

Computer communications and networks have markedly altered our society. The Web, which started as a research project of Sir Tim Berners-Lee in 1989 at CERN has become a global phenomenon. The sprouting of social and technological networks like Facebook, Google+ and the Web has increased and eased long-distance communication. This, in turn, has changed the way we live and work. These inventions were possible thanks to the laying down of communication protocols like IP, TCP and HTTP, routing algorithms (which work hand in glove with graph theory) and wireless networks like Bluetooth.

Processing cores of a supercomputer

 

Distributed computing algorithms which can be applied to cloud computing and concurrent programming (used in email accounts), as well as parallel processing techniques used in supercomputers and large-scale Web servers have helped to cement the place of network technologies in our modern day society. Supercomputers, for example, help us to simulate with high accuracy what the weather conditions within the next week might be. This helps businessmen and transportation companies to revise their plans and avert preventable material and financial losses. The enormous data — structured, semi-structured or unstructured — which is currently to the order of zettabytes (10 to the power of 21 bytes) generated by these networks can in turned be supplied to AI systems as input upon which insights can be drawn from datasets using the principles of statistical analysis, data mining, and appropriate data science tools. Deductions from these datasets are very frequently used by manufacturing companies to tailor the production of their products in such a way as to maximize profits, and by software companies like Google and Facebook, to provide recommended video lists on YouTube and show up lists of friends you may want to add, respectively.

A social graph

 

Let’s round off with computer security, IoT, IoV and quantum computing.

Computer security uses various cryptographic protocols and encryption schemes to provide secure communication and computation. These are necessary to ensure the safeguarding of our digital content distribution, e-voting systems, digital payment systems, and cryptocurrencies. The internet of things (IoT) is concerned with digitizing our physical world with wireless sensors, analyzing the data received and using the results to ameliorate our living standards. IoT has the immense potential to transform our world as it has ushered in the era of smart cities, the industrial internet of things (IIoT), smart homes, wearable IoT devices like Apple watches and connected cars (which will optimize self-driving car technologies).

A glimpse of the IoT

 

The internet of value (IoV) allows for the quick to exchange of goods and services on the internet. It seeks to eliminate middlemen (like banks) in transactions so as to speed up the flow of value. Cryptocurrencies, distributed ledgers and blockchain technology all stem from the vision of building the IoV. Distributed ledgers, in particular, provide a decentralized platform upon which transactions can be carried out without the need of a central authority and thus helps to keep check against manipulation.

Quantum computing, though in its infancy, is a sleeping giant whose coming will instill mixed feelings in the general populace. For all we do know, a quantum computer is capable of breaking all the authentication and encrypting techniques we employ to secure our computer systems. Quantum computing (QC) arose from the marriage of quantum mechanics and computer science in order to leapfrog the processing capacity of present-day chips. QC promises to be a harbinger of technological disruptions, which is the reason why the US and China are injecting millions of dollars of funds into QC research.

A quantum computer chip

And here is the end of our tour. I hope you enjoyed it and learned a great deal about the fields comprised in informatics, their respective applications and the disruptions they have effected. I leave you to reflect on the quote below from Alan Perlis.

— In man-machine symbiosis, it is man who must adjust: The machines can’t. —

A journey worth remembering!

“You are not a failure until you stop trying.” …………………………..Albert Einstein

Albert Einstein, who is widely regarded as the father of Modern Physics, knew exactly what persistence was. After publishing his famous ‘5 papers’ which encapsulated revolutionary ideas in physics like the equivalence of mass and energy and the theory of special relativity in 1905, it took this scientist 10 more years to arrive at his outstanding general theory of relativity. During this decade of inquiry and research, he doubted his own work several times along the way, but he managed to keep on until he arrived at his dream theory, and stood out among his contemporary physicists. Anyway, aside from history, it’s time for me to recount a little bit of my own experience – how having a positive mindset rewarded my efforts to get a good university education.

During my time at secondary school, I had always dreamt of pursuing my university studies at a reputable institution of higher education in Cameroon. I was well aware of the competitive nature of admissions (which are almost entirely based on academic merit) into home-based universities  This motivated me to put more efforts in my high school studies, in order to obtain the required grades. When my physics teacher introduced me to Open Dreams in Lower Sixth (the equivalent of eleventh grade in the US), I was delighted at the prospect of getting a full scholarship to study at a foreign university. From the time I encountered Open Dreams, I never stopped dreaming of obtaining a full scholarship, given that I myself saw that I had the potential to obtain one.

Many a time I ended up with either incomplete college applications, or rejection letters from the universities I had applied to. All of these happened because occasionally I doubted myself and the college application process – especially the latter because it was seemingly overcomplicated. In the meantime, I got into an engineering programme at a home-based university. To be frank, I was not satisfied with the kind of training I received at that university, because it was per se too theoretical. In consonance with what Einstein said: “In the middle of every difficulty lies opportunity”, I decided to turn the tide I faced in my education in my favour. I saw my frustration from my engineering education as a motivation to apply for a fully funded scholarship. I read more and more books and articles on technology, entrepreneurship and solutions to societal issues plaguing the world and Africa in particular. In addition to this, I used my free time at this university to explore what I really loved and wanted to study at the university, as I juxtaposed several degree programmes to see what would be a good fit for me, should I be given the opportunity to get a fresh start into another university.

In time (ultimately God’s time for me), I got the MasterCard Foundation Scholarship to study at the University of Edinburgh. And guess what? I chose to study Computer Science rather than engineering, because after careful thought and examination, I discovered that I wasn’t so enthusiastic to pursue a career in engineering, as opposed to one in IT. Moreover, my passion for mathematics, computation and their respective applications in tech, also led me to switch to my present choice of degree. As I peer closely into the past, I cannot help but sincerely express my heartfelt gratitude to all who supported me throughout the college application process, including but not limited to my family, church, Open Dreams and friends. And I strongly believe my journey into a blissful future has just begun…