Algebra in society

Summary:Algebra provides tools for orderly thinking and problem solving, applicable across a spectrum of pursuits.

Among the many discussions in his 1961 book The Realm of Algebra, science fiction author and biochemist Isaac Asimov described the real-life uses of algebra; explored the role it played in the discoveries of scientists and mathematicians such as Galileo Galilei and Sir Isaac Newton; and suggested the idea that “the real importance of algebra, and of mathematics in general, is not that it has enabled man to solve this problem or that, but that it has given man a new outlook on the universe.” This notion underlies many of the perspectives on algebra in the twenty-first century.

Knowledge of algebra is seen as important not only for scientific research and the workplace but also for teaching general logical thinking and for making decisions that are important to personal well-being and society as a whole. For example, some functional relationships among people’s day-to-day activities that may affect personal decisions include the relationship between how much food a person eats and weight; the amount of exercise and weight loss; and calculations for loans, interest, and other financial matters.

Some would say that the ramifications of these relationships and a lack of understanding of them mathematically are found in the housing crisis of the early twenty-first century and the increase in obesity. Algebra is reported as being a challenging subject for some people.

Many consider algebra to be a major gateway into higher mathematics in both high school and college, and it is thus critical to careers in engineering, science, mathematics, and other disciplines that require advanced mathematics training. Performance of primary and secondary students on algebra tests is one common comparison measure used to evaluate the relative standing of countries with regard to education. Professional organizations like the National Council of Teachers of Mathematics (NCTM) continue to examine the role of algebra in society and make recommendations. Some of the numerous careers that have been cited as requiring algebra include architecture, banking, carpentry, dentistry, civil engineering, nursing, pharmacy, and plumbing.

How Is Algebra Useful?

In 2003, the RAND Corporation’s Mathematics Study Panel underscored the key role of algebra in education by choosing it as one of the panel’s main areas of focus, explaining their decision in part by saying, “Algebra is foundational in all areas of mathematics because it provides the tools (i.e., the language and structure) for representing and analyzing quantitative relationships, for modeling situations, for solving problems, and for stating and proving generalizations.” In algebra, there are general laws or algebraic models that can be used to represent a given scenario.

Algebra is sometimes noted as a type of language that provides answers to all cases at all times and models the relationships between quantities, reducing the need for repeated or inefficient computation. For example, in order to determine the savings in an interest-bearing account after a given period of time, one could compute the savings each month or year by multiplying by the interest rate. However, this computation is cumbersome after many compounding cycles. Instead, the algebraic formula

can be applied directly, where P is the initial investment, r is the interest rate per period, t is the number of periods, and A is the amount of money in the bank after t periods. People may want to know if it is profitable to leave money in a bank subjected to the stated formula. On the other hand, people may want to determine the present and future value of the money they have invested because of the effect of inflation. In other instances, such as taking a car or home loan, similar algebraic laws exist. These laws help people know how much money, for instance, they may save if they pay off their loan earlier than the due date. In the eleventh century, scholar, poet, and mathematician Omar Khayyam explained the following:

…Algebra is a scientific art. The objects with which it deals are absolute numbers and measurable quantities which, though themselves unknown, are related to “things” which are known, whereby the determination of the unknown quantities is possible.… What one searches for in the algebraic art are the relations which lead from the known to the unknown.… The perfection of this art consists in knowledge of the scientific method by which one determines numerical and geometric unknowns.

Early History

Algebra definitions and applications have evolved over time, though many aspects of algebraic thinking and methods that are taught in twenty-first-century schools can be traced back to antiquity. The Babylonians and Egyptians used algebraic techniques to solve problems directly related to the everyday needs of society, such as dividing land and keeping financial records. One such example from Babylonian mathematics is an alternative method for solving cubic equations of the form x³+x²=b, via tabulated numerical values of squares and cubes. The Babylonians were able to solve this polynomial by using the table that gave the values of x³+x² or x²(x+1). They constructed the table to solve: x²(x+1)=1;30 in sexigesimal notation. The “periods” below are used to represent multiplication.

The algorithm used by the Babylonians to find the roots of cubic equations is different from the modern approach, although it can be explained using modern language.

For example, in modern notation, in solving the equation x³ + 2x² - 3136 = 0 set x = 2y. Then the equation can be rewritten as the following:

From the table, y = 7. Since x = 2y, then x = 14.

Topics that are viewed as algebra in contemporary mathematics were often numerical or geometric in nature. The Pythagorean theorem, named for Pythagoras of Samos, can be expressed in terms of the algebraic equation that relates the sum of the sides surrounding a right angle in a triangle squared to the square of the hypotenuse. However, historically, there is evidence that the Babylonians explored numerical versions of the theorem, while the Greeks examined the areas of the geometric squares that sat on the edges of the triangle.

The Pythagorean theorem can be found in twenty-first-century algebra classrooms, and it is useful in setting right angles in constructions and in measuring distance in flat objects. Symbolic notation for algebra was developed in India and became popular in Europe in the seventeenth and eighteenth centuries. Historical methods reflect the unique construction of understanding, indicative of the localized culture at that time. Algebraic methods have also been found in some ancient Chinese works.

Greeks, Hindus, Arabs, Persians, and Europeans all contributed to the development of algebra. The term itself comes from the Arabic word al-jabr, which has several translations including “the science of equations.” The word appears in the title of the early algebra text written by Muhammad Ibn Musa Al-Khowarizmi in the ninth century.

Applied Algebra

For a long time, one major emphasis in algebra was solving polynomial equations, but in the eighteenth century, algebra went through a transformation that broadened the field to include study of other mathematical structures. Around that time, textbooks defined algebra in many different ways. According to mathematician Colin Maclaurin, “Algebra is a general method of computation by certain signs and symbols which have been contrived for this purpose, and found convenient.

It is called an universal arithmetic, and proceeds by operations and rules similar to those in common arithmetic, founded upon the same principles.” Leonhard Euler defined algebra as: “The science which teaches how to determine unknown quantities by means of those that are known.” As the concept of variables was further developed, many physical properties, including time, mass, density, pressure, temperature, charge, and energy, were expressed algebraically.

For instance, Albert Einstein’s equation relates energy to mass times the speed of light squared. In the twenty-first century, defining algebra commonly requires a broader approach. First, one could say that early or elementary algebra is essentially the study of equations and methods for solving them; and second, that modern or abstract algebra is the study of various mathematical structures. High school algebra textbooks typically contain a breadth of topics, such as polynomials and systems of linear equations. These are important in modeling many relationships in society. For example, parabolas represent the paths of ball or bullet trajectories, and systems of linear equations and matrices give rise to digital images. At the college level, students continue their study of algebraic equations in virtually every mathematics and statistics class. Students in a broad range of majors, including the sciences and mathematics, may further their understanding of systems of linear equations and their applications in a linear algebra class.

Mathematics majors in modern or abstract algebra study topics like groups, rings, and fields, and graduate students further explore these and other algebraic structures. These concepts have been useful in chemistry, computer science, cryptography, crystallography, electric circuits, genetics, and physics. Algebra is a core area from the middle grades and high school to undergraduate and graduate mathematics. Research fields include the connections of algebra with other subdisciplines, like algebraic geometry, algebraic topology, or algebraic number theory, and the abstract structures and notions in pure algebra have been applied in many contexts. Some algebraists work for the National Security Agency and others work as professors.

In general, mathematicians and scientists often algebraically derive laws for a given scenario or relationship from patterns. For example, consider a triangle number pattern. It is fairly simple to find the next number recursively but finding larger values such as the 1000th triangular number without a general rule can be more challenging. (See Figure 1 and Table 1.)

Algebra can be used to generalize the preceding case and derive that

so the general law will be

Hermann Weyl noted “The constructs of the mathematical mind are at the same time free and necessary. The individual mathematician feels free to define his notions and set up his axioms as he pleases. But the question is will he get his fellow mathematician interested in the constructs of his imagination. We cannot help the feeling that certain mathematical structures which have evolved through the combined efforts of the mathematical community bear the stamp of a necessity not affected by the accidents of their historical birth. Everybody who looks at the spectacle of modern algebra will be struck by this complementarity of freedom and necessity.”

Many algebraic equations are used in everyday life to meet societal needs. For example, the area of a rectangle is given by the length times the width. There are algebraic equations like finding the area of a square or circle, and also finding volume, which are used in applications like home decorating, cooking, landscaping, and construction. Building houses and fences, determining amounts of material needed for a project, and completing everyday chores use algebra to make work accurate and efficient. Economists use algebraic laws to project business profits or losses and to advise investors and other decision makers. In other instances such as taking a car or home loan similar algebraic laws help people know how much money, for instance, they may save if they pay off their loan earlier than the due date.

Many formulas are easy to use and can easily be entered in a hand calculator or computer to generate the required result. Such formulas have been adapted to Web-based applets and software like spreadsheets to track financial records, making them widely accessible and often easy to use.

Mathematician Roger Cooke explained, “Algebra provided more than just a compact notation for writing down relations among variables. Its rules made it possible to manipulate those laws on paper and derive some of them from others. For example, a consequence of Kepler’s third law is that the ratio…of the square of a planet’s period to the cube of its distance from the sun is the same for all planets…Kepler’s third law and Newton’s law of gravitation are equivalent statements, given certain basic facts of mechanics.” Kepler’s laws were named for Johannes Kepler and Newton’s for Sir Isaac Newton. The ability to express algebraic relationships using variables and rates of change in calculus increased the applicability of equations in a wide variety of contexts.

The U.S. Bureau of Labor Statistics highlights the importance of coursework in algebra for numerous careers, including brickmasons, blockmasons, and stonemasons; carpenters; computer control programmers and operators; construction and building inspectors; engineers and engineering technicians; line installers and repairers; machine setters, operators and tenders in metal and plastic; machinists; opticians; physical therapist assistants; power plant operators, distributors and dispatchers; sheet metal workers; surveyors, cartographers, photogrammetrists, and surveying and mapping technicians; radiation therapists; tool and die makers; and veterinarians.

Algebra’s Role as a Gateway

Some would argue that in the United States, mathematics achievement has not met the same standards of excellence as in other developed countries, and that, as a result, students may not be prepared to enter college. Some historians trace the growing need for mathematics education to the turn of the twentieth century or the Industrial Revolution, when there were debates about the appropriate level of mathematics for high school education . Historically, popular opinion was often against algebra as a subject of widespread study in secondary schools, since many did not see clear connections between algebra and real-world needs. Mathematics educator W. D. Reeve cited one newspaper editorial as an example of such an attitude in a 1936 National Council of Teachers of Mathematics report, The Place of Mathematics in Modern Education, saying the following:

Quite frankly, I see no use for algebra except for the few who will follow engineering and technical lines.… I cannot see that algebra contributes one iota to a young person’s health or one grain of inspiration to his spirit.… I can see no use for it in the home as an aid to a parent, a citizen, a producer, or a consumer.”

The same report noted deficits in algebra skills even among graduate students and relatively high failure rates for algebra students in some high schools, such as in New York City, which were used by some as additional arguments against algebra’s broad inclusion in the high school curriculum. With regard to who should and should not study algebra, Reeve countered: “. . . no one, I think, has the wisdom to decide who will profit most by its study or predict who the future Newtons and Einsteins are to be.”

Mathematician and philosopher Alfred Whitehead stated the following:

Quadratic equations are part of algebra, and algebra is the intellectual instrument which has been created for rendering clear the quantitative aspects of the world. There is no getting out of it. Through and through the world is infected with quantity. To talk sense, is to talk in quantities. It is no use saying that the nation is large.… How large? It is no use saying that radium is scarce.… How scarce? You cannot evade quantity. You may fly to poetry and to music, and quantity and number will face you in your rhythms and your octaves.… This question of the degeneration of algebra into gibberish, both in word and in fact, affords a pathetic instance of the uselessness of reforming educational schedules without a clear conception of the attributes which you wish to evoke in the living minds of the children.… First, you must make up your mind as to those quantitative aspects of the world which are simple enough to be introduced into general education; then a schedule of algebra should be framed which will about find its exemplification in these applications.

Other newspapers like the Columbus Dispatch supported broad high school mathematics education during that time period, asserting that schools should provide the “mathematical key” to the “gateways of a larger life.”

Algebra eventually became commonplace in high schools and some middle schools, with basic algebraic concepts often introduced even in the primary grades, and yet questions about how to teach algebra continued. Algebra is usually a prerequisite for all higher mathematics courses in both high school and college, and, in some cases, it is required for high school graduation. Students will not advance in many majors or career paths unless they pass algebra, and the result is that some students change majors or abandon education altogether. Students requiring remediation courses at the college level are fairly common.

The result is that in the twenty-first century, algebra is still viewed by many as a major gatekeeper to educational and career advancement, and learning algebra has been promoted as a civil rights issue for every U.S. citizen, though many of the same arguments from past decades continue to be debated. In the latter twentieth century, algebra education became a renewed topic of discussion from local school districts all the way to the White House. The RAND Corporation’s panel further explained its decision to focus on algebra by saying, “Without proficiency in algebra, students cannot access a full range of educational and career options, and this curtailment of opportunities often falls most directly on groups that are already disadvantaged.”

At the same time, naysayers continue to publish counterpoints regarding algebra’s lack of utility. One 2006 Washington Post article about a student named Gabriella, who purportedly dropped out of high school after failing her algebra course many times, asserted that writing teaches logical reasoning more effectively than algebra and stated that many students will “never need to know algebra” in the real world, since most mathematics can now be done by computer or calculator. It concluded that having an algebra requirement for high school graduation is potentially more detrimental than helpful because it may spur students to drop out who otherwise might have graduated.

This article spurred many further discussions, and it appeared to reflect the author’s own difficult experiences with algebra, a phenomenon that has been reported by many educational researchers and that drives further curricular revisions. Authors of algebra textbooks and self-help books have explored different ways to help students connect to algebra. For instance, actress Danica McKellar has written algebra readiness and algebra books that include stories and characters in order to express equations and solutions in contextual situations. Some educators incorporate mnemonics, songs, or other memory techniques such as First, Outside, Inside, Last (FOIL) in order to teach the multiplication of two binomials. Other authors highlight real-life applications, historical connections, or solutions using technology.

Many national reports have indicated that education in the United States is in a critical period, and some would say particularly in mathematics and science. Educators and politicians have proposed changes to the mathematics education curriculum to prepare U.S. students. The number of students entering college and requiring courses that enable them to be effective in the workplace is rising. Further, engineering and other technical fields that were once seen as elite or remote are increasingly a part of daily life, including computing, electronics, business, and architecture. Technology is changing every day, which has changed society, including mathematics. As a result, there is an increased need for people who can adapt to the changes and continue being effective in society. In this context, there has been a movement to reform algebra education so that it can be more readily accessed by everyone. The “algebra for all” movement has been a central point within the reform initiatives. National standards such as those published by the NCTM have stressed the need to make algebra more accessible to students, and they often outline both the content to be covered and instruction expectations. Some research has shown that students who take algebra by eighth or ninth grade are more likely to pursue higher mathematics, though this cannot be interpreted as a cause-and-effect relationship.

Bibliography

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Gallian, Joseph. Contemporary Abstract Algebra. 7th ed. Belmont, CA: Brooks Cole, 2009.

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