Problem solving in society

SUMMARY: Mathematics is used to find and solve problems, often spurring new mathematical investigations.

Problem solving is fundamental not only to the learning and application of mathematics as a student, but to all walks of life. Many people consider mathematics and problem solving synonymous. However, there are many mathematicians who do not solve problems or who do more than solve problems. Some work to build new theories or advance the language of mathematics. Others unify or explain previous results, sometimes from many fields of mathematics. Yet others consider the very nature and philosophy of mathematics as a discipline. In twenty-first-century society, mathematics teaching at all levels seeks to develop students’ abilities to effectively address a wide variety of mathematics problems, including proving theorems; reducing new problems to previously solved problems; formulating and solving both real-life and abstract word problems; finding and creating patterns; interpreting figures, graphs, and data; developing geometric constructions; and doing appropriate computations or simulations, often with computers or calculators.

Problem solving is also an instructional approach in which students actively learn fundamental concepts through their contextualization within problems rather than from a passive lecture. What fundamentally connects these activities, beyond the mathematics techniques and skills necessary to solve them, is the framework of “how to think.” Students must have the necessary tools and techniques at their disposal through a solid education in the fundamentals. They must also be able to either analyze the characteristics and requirements of a problem in order to decide which tools to apply, or know that they do not have the appropriate tool at their disposal. Further, students must practice with these mathematical tools in order to become skilled and flexible problem solvers, in the same way that athletes or craftsmen practice their trades. As Hungarian mathematician George Pólya expressed, “If you wish to become a problem solver, you have to solve problems.” This idea extends to the notion that problem solving is by its nature cyclic and dynamic. In many cases, the solution to a problem results in one or more new problems or opens the path to solving an older problem for which a solution has previously proven elusive. Sometimes, mathematics problems have real and immediate applications, and many new mathematical disciplines, like operations research or statistical quality control, have developed from these sorts of problems. In contrast, there are many issues in theoretical mathematics that do not appear to have any immediate benefit to society. In some cases, people question the need to explore such abstract problems when there are more immediate needs. Often, these abstract problems turn out to have very concrete applications decades or even centuries after their initial introduction. Even if that is not the case, theoretical problem solving adds to the growing body of mathematics knowledge and, just as importantly, shows people yet another way to think about the world.

History

The mathematics body of knowledge is not static; it has been evolving with humans. As soon as humans organized themselves into communities attached to the land, benefits rapidly emerged. Certainly, an advantage was an increase in agricultural and livestock productivity. As a result, part of the harvest and the cattle was accumulated for worse times. Accumulation demanded certain mechanisms to identify the ownership and use of the land (the process of land surveying) and to record who contributed to what was collected (the system of counting). The success of such social structure allowed skilled individuals to take advantage of their abilities to exchange the resultant products for food surplus (the beginnings of commerce). The development of commerce demanded a new tool to register the commercial operations in order to recognize who was implicated and the amount involved. This tool was based in a new kind of language (mathematics) able to do operations such as additions, subtractions, iterative sums, and partitions that natural languages were unable to support. As with any language, it consisted of two elements: notation to represent ideas (numbers) and syntax to manipulate these ideas (calculation).

After the accumulation of goods came the capability to organize collective efforts. It was possible to build massive public works. Warehouses, markets, fortresses, temples, aqueducts, and even pyramids were constructed in urban centers and their surroundings. Construction presented a new problem related to the manipulation and combination of forms. Early exercises were based on rules used for land surveying; for instance, to calculate areas and volumes. Additional difficulties arose when public works increased their complexity; hence, the application of forms and their interactions to develop better habitats gave rise to the development of architecture as an independent discipline. The Greeks separated land surveying from the study of spatial relations and forms; as a result, geometry was born. This discipline was used to solve abstract mathematical problems. For instance, Pythagoras recognized the relation between the sides of a right triangle as a²+b²=c² (the Pythagorean Theorem), and Archimedes studied the relation between the circle’s circumference and its diameter. The latter is known as pi (π), an irrational number with the value of 3.141592653589793238462643383279502… .

The Problem of Representation and the Dynamics of Change

With the accelerated increase of richness and variety in social interactions, intractable problems of representation appeared. Operations were required to record social experiences from an ever growing dynamism. This endeavor made limitations in the notation systems available at that moment evident. Hindis and, afterwards, Arabs and Muslims developed the positional decimal system still in use in the twenty-first century. The decimal system allows the representation of arithmetic operations without the need to use an abacus. Changes in quantities demanded introducing a general notation for variable and constant amounts, which were linked by operators to form different sentences, called “equations.” The study of these relations is known as “algebra.”

The capability to represent abstract ideas and their relations allowed mathematicians at the beginning of the sixteenth century to discuss problems related to the dynamics of change. In fact, the field of astronomy proposed new challenges to mathematics. Between 1507 and 1532, Copernicus presented a series of works where he substituted the traditional viewpoint, which located the Earth at the center of the universe (the geocentric view), with another where the sun was at the focus (the heliocentric view). This view helped to explain inconsistencies in the stellar movement, such as the retrograde displacement of planets. Around 1605, Johannes Kepler empirically discovered the elliptic orbit of planets around the sun. He also noticed that the line that joins each planet with the sun (called the “radius vector”) sweeps the same area in the same period of time. Galileo focused his telescope to Jupiter, and, in 1610, he posited that the lights surrounding that planet were, in fact, satellites. To demonstrate all of this in mathematical terms demanded the study of change in relation to time, something impossible to solve at that moment. Isaac Newton and Gottfried Leibnitz simultaneously developed a useful procedure known as “calculus.” When it is used to represent the change of a certain quantity in relation to another in terms of infinitesimal moments, it is called “differential calculus.” Interestingly enough, this procedure can be reversed to reckon space sections bounded by different functions. The general procedure consists on dividing them into additive infinitesimal blocks—a process named “integral calculus.” Both procedures operate in an inverse manner through the fundamental theorem of calculus.

The Problem of Estimation

In the seventeenth century, additional problems appeared when the practical world confronted an impossible question. How can one characterize something that is not stable enough to be counted? For instance, in order to establish public policies, politicians need to know what resources are at their disposal—the demographic and economic capabilities, which can be determined in a census. The main problem with exhaustive counting of populations is that they change. There are births and deaths. In order to solve this issue, one method is to select a fraction (called a “sample”) of the object of study (called the “population”), to identify the sample characteristics and to generalize them to the population. Advantages for this sampling procedure are lower costs and faster data collection than following a comprehensive census. But there is an important difficulty: how to guarantee that the characteristics of the sample are the same as those of the entire population. One needs to estimate the sampling error because of selecting a sample that does not represent the population and to define a confidence interval by identifying the reliability of the estimate. The part of mathematics interested in this kind of problems is known as “statistics.”

Statistics helps to solve many technical problems. Statisticians may need to (1) estimate the size of a population, as Laplace did in 1786 for France, by using a sample; (2) describe a population in terms of different numerical relations, such as its expected value (called “average”), its most frequent value (called “mode”), the limits of the data series (called “range”), the value that separates the higher half of the data series from the lower half (called “median”), and the data dispersion (called “standard deviation”); (3) test a hypothesis as J. H. Jagger did in 1873 at the Beaux-Arts Casino at Monte Carlo, when he collected results from a roulette wheel to prove that it was fraudulent; (4) estimate if a process needs products of a certain quality or it requires to be fixed, as in statistical quality control; (5) identify if changes in a process result in a positive outcome (called “correlation”), such as the Hawthorne study done in a working line to correlate the increase in illumination with workers’ productivity; (5) predict and forecast future outcomes by means of recognizing patterns of behavior, what is known as “regression”; (6) extrapolate future data through the analysis of previous results; (7) reconstruct incomplete series data by means of that which is known and available, through interpolation; or (8) model the behavior of an entity in order to transform data into valuable information (called “data mining”).

The Problem of Decision Making

The Industrial Revolution introduced a massive change in the social order. Early stages of the period witnessed the substitution of agricultural workers with machines by the thousands. It represented an increase in the productivity for many industries and services, mainly textiles and transportation, to levels never before seen. It surpassed the previous cumulative capacity of mankind. It also implied a surplus of energy with the use of internal combustion engines and electrical power generation. However, finding the equilibrium in this new social order was not an easy endeavor. Two world wars witnessed this planetary enterprise, and the postwar era during two different visions of the best way to organize the global society developed into a mortal conflict: capitalism versus communism.

At the beginning, the Industrial Revolution promised benefits with no end, although it made the medieval work system based on guilds inoperative. Groups of artisans loyal to a closed system of hierarchical progression were substituted by interchangeable clusters of men and machines located at industrial centers with short-term economic success as its main performance criterion. These were operationalized in terms of effectiveness and efficiency, and optimization was the prime improving activity. Methods based on empiricism and not on tradition acquired a new value. For instance, in 1840, Charles Babbage realized a study about mail classification and transportation; the result was the institution of the Uniform Penny Post; a taxation procedure by which a letter not exceeding half an ounce in weight could be sent from any part of the United Kingdom to any other part for one penny. In 1911, Taylor proposed a series of managerial principles that were the foundations of what is currently known as “management science” or “operational research.” This Science of the Better consists in the application of advanced analytical methods to help make better decisions.

Operational Research took shape just prior to World War II. At the beginning, exercises were focused on solving problems of fighter direction and control in the British air defense system. The new radar system acted as an early warning system that was able to identify German aircraft before they would bomb air bases, ports, industrial areas, and cities. Success demanded, later during the war, to extend these exercises to the Atlantic Ocean. Massive ship losses because of the attacks of U-boats (German submarines) put Allied supplies to Europe and North Africa at risk. Accordingly, different analyses were conducted to increase the U-boat sinking rate. Different criteria were mathematically explored and solutions were implemented, including (1) identifying which kind of aircraft was the best suited to chase German submarines; (2) reckoning the time at which depth charges should explode, and (3) defining the size of merchant fleets that minimizes Allied losses when crossing the Atlantic.

From the success of analyzing the performance of military operations, this field of mathematics was extended to other industrial and social activities. Many different problems have been studied and alleviated by this approach, including (1) community development, in order to organize collectives, support strategies that deal with social dissatisfactions, help groups in rural communities and developing countries, and create the social conditions for effective public policies; (2) criminal justice, to maintain a safe society by optimizing the use of resources allocation that enforce the law and reduce spaces for organized crime and to assess policy impact; (3) education, to evaluate teaching quality, students learning experiences, and assessment procedures; (4) efficiency and productivity analysis; (5) healthcare services; (6) logistics and supply chains; (7) quality control; (8) security and defense; (9) scheduling; (10) strategic management; and (11) transport.

The Problem of Prediction in a Complex World

The acquisition, distribution, and use of knowledge are key factors for the development of individuals and society, an idea that has shifted social structures to more complex levels of organization. The introduction of concepts such as “entrepreneurship” (a wild spirit who causes creative destruction by innovation and disruption) or “leadership” (a process of social influence and emotional contagion) are the result of recognizing that people’s actions affect many others in non-evident ways. Economy, ecology, management, and politics require new approaches as these phenomena develop with intensities never before expected. The limitation of resources demands humans to use them responsibly and to make decisions for a better future. The main difficulty consists of predicting the future from the present. How can a person predict future consequences of actions to recognize good actions from bad ones?

Advising people on how to act is an age-old business. For a long time, the unique sources at disposal were divinely inspired or supported by powerful collectives. However, since the 1800s, the emphasis shifted toward scientific study of the environment regarding which actions take place. Prediction was focused on learning from the past and expecting the future to behave similarly, what is known as “time-series procedures.” These can be useful where individual decisions have little impact on the overall behavior; for example, the results of the lottery or the weather conditions for the next few days.

Accordingly, different patterns can be found in the data (such as horizontal, seasonal, cyclic, or trend), but no explanations for the phenomenon under study have been developed. Explanatory models require assuming a relationship between what one wants to forecast (called the “dependent variable”) and something one knows or controls (called the “independent variable”). Through a regression analysis one may minimize differences between observations and the points from an expected trend, linear or not, which can be adjusted to indicate certain seasonality. For more complex phenomena, one may introduce additional independent variables in order to conduct multiple regression analysis. In certain conditions, this approach enhances information for a better decision-making process but assumes the non-evolutionary viewpoint that the best model for the future is the one which better fits historical data. This approach also reduces the size of phenomena under scrutiny because modeling a real complex phenomenon such as the world’s climate goes easily beyond twenty-first-century computers’ capabilities and human understanding.

Complexity is related to many things such as size, difficulty, variety, order, or disorder. However, it has nothing to do with complication. Anything complicated can be solved, usually by introducing more resources to crack current problems. Conversely, complexity is associated with the impossibility of guaranteeing future behaviors based on current ones. The mathematical treatment of complexity introduced a discipline known as “chaos theory.” It is a collection of mathematical, numerical, and geometrical techniques that allow mathematicians to deal with non-linear problems that do not have explicit general solutions. It is based in the use of differential equations to analyze dynamic behaviors extremely sensitive to initial conditions. In this context, predicting the future has to do with recognizing stable equilibrium points (called “fixed point attractors”), those that appear when dynamic systems stop. An attractor indicates the natural tendency of a system to behave in a certain way in the long-term future, if nothing else disturbs it. Common physical examples of this kind of behavior are pendulums and springs. Attractors are used for decision making in different fields, such as finance, where investors try to identify stock market tendencies. Some major applications related to its origins are weather prediction, solar weather prediction models, and predicting fisheries dynamics.

The increase of computing power allows mathematicians to run mathematical models based in little pieces of code that represent specific behaviors (called “intelligent agents”). Agent-based models can be used to study complex behaviors to simulate individual behaviors, such as people’s movements inside stadiums or automobiles avoiding traffic jams. Other studies related to self-organized and self-organizing behaviors can also be conducted as they can represent phenomena from economy and financial markets; opinion dynamics; emergency of social rules and institutions; creation or disappearance of companies; and technology innovation, adoption, and diffusion.

To recognize stability areas and patterns in complex behaviors resulting from a multiplicity of agents interacting is then at the basis of the next social challenge, and procedures to deal with this are at the edge of twenty-first-century capabilities. The study of elements and their interactions have developed new viewpoints to observe reality. To visualize problems as a myriad of elements richly interconnected with unseen behaviors and consequences has introduced notions such as “systems” and “networks” in discourse. In 1950, Ludwig von Bertalanffy, a biologist, recognized similar fundamental conceptions in different disciplines of science, irrespective of the object of study. He tried to represent those rules through a language to describe such entities, which he named the “General System Theory.”

A year before, Werner introduced the notion of communicative control in machines and living beings by looking at the effects of feedback on future behaviors. He named it “cybernetics.” Based on this, in 1956, Ashby provided a single vocabulary and a single set of concepts suitable for representing the most diverse types of systems. Since then, different researchers have developed alternative methodologies to describe phenomena not in terms of problems and solutions, but in terms of satisfaction and alleviation. This has been used to deal with non-technical problems—those considered impossible to solve only through analytical tools, as they include humans’ interactions. In this context, relations between individuals are diagrammed and studied in terms of bunches of nodes interconnected by links. From this viewpoint the image of a “network” emerges. This notion has been developed, for instance to measure the “distance” between two persons from different places and contexts and reckoned that the average number of intermediate people between them is 5.5, hence the phrase “six degrees of separation.” Network analysis is important as it can be used to model and study phenomena such as the Internet and its vulnerability to hackers, viruses and their uncontrollable expansion, or technology innovation and its diffusion. Future developments on this area are expected.

Bibliography

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