Algebra The Language of Science by James Nickel

Summary • • The misdirection of High School Mathematics Education. Biblical Christian direction (or

Acknowledgement • I am grateful to Morris Kline’s Why the Professor Can’t Teach: Mathematics and the Dilemma of University Education (St. Martin’s Press, 1977) for much of the following analysis. Copyright 2008 www. biblicalchristianworldview. net

20 th Century High School Mathematics: Scope and Sequence • • 9 th grade: Algebra I 10 th grade: Geometry 11 th grade: Algebra II 12 th grade: Pre-calculus, Calculus, or Solid Geometry Copyright 2008 www. biblicalchristianworldview. net

Analysis • • • Note: As I analyze this Scope and Sequence, I am going to play the “devil’s advocate. ” I am going to be somewhat cynical with a purpose in mind. Let’s see if we can view this material from the eyes of the typical student; i. e. , what I experienced in the 1960 s. Copyright 2008 www. biblicalchristianworldview. net

th 9 • • Grade: Algebra I Almost entirely a series of disconnected, atomized processes, a mélange of topics. Note, in the early 21 st century, there is a move to cover Algebra I in the 8 th grade. Copyright 2008 www. biblicalchristianworldview. net

Algebra I Topics • • • Factoring. Operations with polynomials such as x 2 + 9 x + 20. Operations with fractions such as Laws of exponents. Solution of equations of various degrees and with one or more unknowns. Copyright 2008 www. biblicalchristianworldview. net

Algebra Perspectives • • • Algebra is a means to something else, not the end in itself. Algebra is the spelling, grammar, and rhetoric of most mathematics, but it is not literature in and of itself. This maze of letters that represent number symbols that represent numbers was fully developed in the 17 th century. Copyright 2008 www. biblicalchristianworldview. net

Two Levels of Abstraction • • • Level 1: Numerals are symbols of number. Level 2: Letters are symbols of numerals. For example, instead of 3 x = 5, we have ax = b. Copyright 2008 www. biblicalchristianworldview. net

Two Levels of Abstraction • • • But, generalizations about men and women are meaningful only to those who have had lots of experience with individual men and women. The analogue for Algebra is lots of experience with various manipulations of numbers. Unfortunately, few students have more than a nodding acquaintance with arithmetic at the time they are propelled into Algebra. Copyright 2008 www. biblicalchristianworldview. net

th 10 • • Grade: Geometry The Geometry study is Euclidean or Plane (two dimensions). Here, the student is confronted with dozens of theorems all proved in a logical sequence and, to most students, the goal seems to be to prove as many as possible. Copyright 2008 www. biblicalchristianworldview. net

th 10 • • Grade: Geometry Students are confined to learning inflexible and obtuse trains of thought that permit no derailment. Just why anyone wants these theorems and how these theorems and their proofs were ever conceived is not treated. Copyright 2008 www. biblicalchristianworldview. net

th 11 • • • Grade: Algebra II Repeats the 9 th grade subject matter because students did not learn that material. Adds more algebraic processes. Introduces trigonometry, a subject in which, among other topics, the students learn many identities at the risk of losing their own. Copyright 2008 www. biblicalchristianworldview. net

th 12 • • Grade: Various Schemes Coursework has not been stable. But the material, whether solid geometry or the beginnings of calculus, is taught dogmatically and has been no more enlightening or enthralling than the preceding subjects. Copyright 2008 www. biblicalchristianworldview. net

Who is Responsible for this? • • Who generated this curriculum? It was developed in the 19 th century by university professors. Before this time, these high school subjects were only taught in college. They gradually filtered down into high school keeping the same college content. Copyright 2008 www. biblicalchristianworldview. net

Rationale for this Program • • Utility in daily life. Value of mental discipline; i. e. , mathematics teaches you how to think. Learning of higher, nobler truths; i. e. , a 2 – b 2 = (a + b)(a – b) would uplift any soul. Preparation for college and professional life. Copyright 2008 www. biblicalchristianworldview. net

State of Things since WWII • • 1950 s: the ineffectiveness of this traditional curriculum (made evident by the needs of WWII) stimulated some university professors to reform the process. This reform resulted in the “new math” of the 1960 s. Copyright 2008 www. biblicalchristianworldview. net

1960 s New Math • • Emphasis on theory, generalization, rigor, and structure of

1960 s New Math • • These “innovations” may have ensured the stability of the several mathematical edifices, but they also pulled the students so far down into the dark earth that they could no longer see the surface. e. g. , 2 + 3 = 3 + 2, but does 2 + 3 = 5? It is now been recognized (in hind-sight) that this reform was not an improvement; it was a disaster. Copyright 2008 www. biblicalchristianworldview. net

Summary • • The contents of all four years, as presented, are abstract, dull, boring, and intrinsically meaningless. It consists, at best (to use Alfred North Whitehead’s phrase), of inert ideas. Copyright 2008 www. biblicalchristianworldview. net

Analysis of Rationale 1. 2. 3. 4. Utility in daily life. Value of mental discipline. Learning of higher, nobler truths. Preparation for college and professional life. Copyright 2008 www. biblicalchristianworldview. net

Utility • • • Does the average educated person use this knowledge in daily life? Do even mathematics teachers, who know the subject, ever use the quadratic formula, the Pythagorean Theorem, or the trigonometric identities outside of the classroom? The honest answer is no. Copyright 2008 www. biblicalchristianworldview. net

Utility • • • Of course, the students learn that a straight line is the shortest distance between two points. That is useful. But even a donkey knows that: put some food at a distance from him and watch the path he takes. Copyright 2008 www. biblicalchristianworldview. net

Utility • • • The subject matter of high school mathematics–that is, the subject matter per se–is worthless knowledge. One or two topics, such as the calculation of compound interest, may prove useful. But the exceptions do not alter the general assessment. Copyright 2008 www. biblicalchristianworldview. net

Thinking • • High school courses have not taught thinking, pedagogue’s assertions to the contrary. The traditional Algebra teaches memorization of processes. Copyright 2008 www. biblicalchristianworldview. net

Thinking • • Geometry purportedly emphasizes proof and therefore thinking. However, because the proofs are arranged in a logical sequence and this sequence is not natural, the rationale eludes the students and they are forced to memorize here, too. Copyright 2008 www. biblicalchristianworldview. net

Thinking • The students’ appreciation of proof in geometry is epitomized in their oftrepeated remark that geometry is where you make proofs in two columns. Copyright 2008 www. biblicalchristianworldview. net

Nobler Truths • All the preaching and rhapsodizing will not ennoble such ugly ducklings

Preparation • • Of those who go to college, at most 10% will need the techniques and theorems now taught in the high schools. The others either take no mathematics in college–many colleges no longer require a mathematics course for the bachelor’s degree–or take a liberal arts course that does not use most of the currently taught high school mathematics. Copyright 2008 www. biblicalchristianworldview. net

Preparation • • Note the growing number of “mathematical development” (i. e. , remedial math) courses in most 2 -year and 4 -year colleges even the basics are not being learned by many high school students. About 5% to 15% of high school graduates will use mathematics professionally. Copyright 2008 www. biblicalchristianworldview. net

What to do? • • • Don’t teach any math in high school. And

The Purpose of this Cynicism • These arguments against the teaching of high school mathematics are arguments against the framework: – – The kind of mathematics that has been taught. The justifications traditionally and currently given. Copyright 2008 www. biblicalchristianworldview. net

Reformation in Framework • The Bible gives direction for viewing reality. Copyright 2008 www.

Reformation in Framework • • • The infinite, personal, and Triune God made the heavens the earth and they reflect a rational order based upon His Wisdom. Man, made in God’s image, can thus reason with this order. Operational science is man reasoning with this order in terms of observation and experimentation. Copyright 2008 www. biblicalchristianworldview. net

Reformation in Framework • • • Biblical Christianity is a faith that is grounded in history. God commands man, male and female, to cultivate the garden; to develop the implications of creation relationally. Hence, history and culture are important pieces of the framework. Copyright 2008 www. biblicalchristianworldview. net

Implications for Algebra • Place the instruction of Algebra in the context of history,

Biblical Christian Education • • • What are the objectives of a Biblical Christian high school education? The courses in literature, history, science, economics, and foreign languages are intended to enable future adults to live more insightfully, wisely, and enjoyably. In short, they are an introduction to our culture and the role of Biblical Christianity therein. Copyright 2008 www. biblicalchristianworldview. net

Implications for Mathematics • • The mathematics courses should serve the same objectives. Hence, the mathematics we teach should be worth knowing for the rest of the lives of all students. Copyright 2008 www. biblicalchristianworldview. net

Implications for Mathematics • • Mathematics courses should contribute to a truly liberal arts (freedom in Christ) education wherein students get to know not only what the subject is about but also what role it plays in our culture and our society. We must teach not just what mathematics is but what it does as a Biblical Christian “tool of dominion. ” Copyright 2008 www. biblicalchristianworldview. net

The Arabic Connection • • • Algebra is an Arabic word. In the 7 th century AD, the followers of Mohammed began ransacking the nations of North Africa, the Middle East, and Europe. In spite of some destruction of knowledge and books, Muslim leaders generally respected and absorbed the manuscripts of the ancient Greek mathematicians. Copyright 2008 www. biblicalchristianworldview. net

The Hindu Connection • The Arabs were also responsible for introducing to Europe several valuable concepts that they borrowed from the Hindus of India: – – The base 10 decimal system. Positional notation. The number zero. Negative numbers. Copyright 2008 www. biblicalchristianworldview. net

The Decimal System • • The decimal system of writing numbers (e. g. , 1, 2, 3, 4, etc. ) proved to be much more efficient than the old, cumbersome Greek letters or Roman numerals. This method of number writing acted as an indispensable asset in later developments of European mathematics and science. Copyright 2008 www. biblicalchristianworldview. net

Algebra: Meaning • In 830 AD, the astronomer Mohammed ibn Musa al‑Khowarizmi (ca. 780 -ca. 850), working in Baghdad, wrote a book titled Hisab al‑jabr w’al-muqabalah, which literally means “science of reunion and opposition” or more freely “science of transposition and cancellation. ” Copyright 2008 www. biblicalchristianworldview. net

Algebra: Meaning • • • Al-jabr, used in a non-mathematical sense, was something a barber did in those times. A barber not only cut hair, he mended bones. Thus, an algebrista was a bonesetter (chiropractor). Copyright 2008 www. biblicalchristianworldview. net

Algebra: Meaning • • In Galatians 6: 2, the Apostle Paul states, “Brethren, if a man be overtaken in a fault, ye which are spiritual restore such a one in the spirit of meekness, considering thyself lest thou also be tempted. ” The Greek word for restore means “to join back together” (as in setting a broken bone) or “to mend” (as in repairing torn nets). Copyright 2008 www. biblicalchristianworldview. net

Algebra: Meaning • • The Greek word for restore means exactly the same as the Arabic word algebrista. In a mathematical sense, solving an equation (determining values of a variable that make the equation true) is like restoring, mending, or resolving it. Copyright 2008 www. biblicalchristianworldview. net

Transition to Europe • • This Arabic text became known in Europe through Latin translations and was shortened to the one word, “algebra, ” which became synonymous with the science of equations. Arabic mathematicians did not use simple symbols like the Symbolic Algebra we used today (e. g. , x, y, etc. ); they used words, or at best, abbreviations of words (syncopated Algebra). Copyright 2008 www. biblicalchristianworldview. net

Transition to Europe • The decimal system needed Simon Stevin (1548 -1620) and François Vieta (1540 -1603) to acquire the conceptual precision demanded by systematic scientific work. Copyright 2008 www. biblicalchristianworldview. net

Soul of Science • • The very soul of science consists in theoretical generalizations leading to the formulation of quantitative laws and systems of laws. India had practicality, craftsmanship and organizational talent, but science in the above sense did not exist in India prior to its invasion by Alexander the Great. Copyright 2008 www. biblicalchristianworldview. net

Soul of Science • Tobias Dantzig (1884 -1956), “Algebra. . . enables one to transform literal expressions and thus to paraphrase any statement into a number of equivalent forms. . . it is this power of transformation that lifts algebra above the level of a convenient shorthand. . . ” Copyright 2008 www. biblicalchristianworldview. net

Soul of Science • “the literal notation made it possible to pass from the individual to the collective, from the ‘some’ to the ‘any’ and the ‘all’. . . It is that made possible the general theory of functions, which is the basis of all applied mathematics. ” Copyright 2008 www. biblicalchristianworldview. net

Soul of Science • • One and the Many: Biblical view of the world enabled man to harness the incredible power of Algebra. Robin G. Collingwood (1889 -1943), “The possibility of an applied mathematics is an expression, in terms of natural science, of the Christian belief that nature is the creation of an omnipotent God. ” Copyright 2008 www. biblicalchristianworldview. net

Culture and Science • • Christ is the Lord and Savior of both. Hence,

Gate and Key of Science • English philosopher and scientist Roger Bacon (ca. 1214 - ca. 1294), nicknamed the “Admirable Doctor, ” said in his Opus Majus (published in 1267), Part 4: “Mathematics is the gate and key of the sciences…. Neglect of mathematics works injury to all knowledge, since he who is ignorant of it cannot know the other sciences or things of this world. ” Copyright 2008 www. biblicalchristianworldview. net

Gate and Key of Mathematics • Algebra, for it prepares us not only to understand the sciences, but also such subjects as medicine, art, statistics, and economics. Copyright 2008 www. biblicalchristianworldview. net

Christ: the Gate and Key of Algebra • From the perspective of the Biblical Christian worldview, learning Algebra is important to our worship because Algebra is the language of science, science is the study of what God in Christ has made, and what God in Christ has made is an unveiling of His glory. Copyright 2008 www. biblicalchristianworldview. net

What History Teaches • • It is beyond dispute that, in history, mathematicians were inspired by real (i. e. , physical and scientific) problems and found the meaning of mathematics through them. Equally beyond dispute is that applications to real problems are a pedagogical necessity. Copyright 2008 www. biblicalchristianworldview. net

Pedagogical Implications • • However, there is some question about which applications will be interesting and meaningful to students. Only experience will enable teachers to determine the best choices. Copyright 2008 www. biblicalchristianworldview. net

Pedagogical Implications • • Fortunately, the choices are so numerous and so varied, and interest in the real world is so much more widespread than interest in abstract mathematics, that attractive applications can surely be found. At the very least, it is easier to arouse curiosity about real problems than about mathematics. Copyright 2008 www. biblicalchristianworldview. net

Pedagogical Implications • The use of real and, especially, physical problems serves not only to answer the question of what value mathematics has but also gives meaning to it. Copyright 2008 www. biblicalchristianworldview. net

Examples • • Negative numbers are not just inverses to positive integers under addition; they are the number of degrees below zero on a thermometer. The ellipse is not just a peculiar curve; it is the path of planets and comets. Copyright 2008 www. biblicalchristianworldview. net

Examples • Functions are not sets of ordered pairs; they are relationships between real variables such as the height and time of flight of a ball thrown into the air, the distance of a planet from the sun at various times of the year, and the population of a country over some period of years. Copyright 2008 www. biblicalchristianworldview. net

Implications • To rob mathematical concepts of their meaning is to keep the branches

Proposal • The proposal that the first two or three years of secondary mathematics contain all sorts of genuine applications may seem radical, but a little perspective may correct this impression. Copyright 2008 www. biblicalchristianworldview. net

What History Teaches • In the 17 th century, mathematics courses comprised astronomy, music, surveying, measurement, perspective drawing, the design of optical instruments, architecture, and the design of fortifications and machines. Copyright 2008 www. biblicalchristianworldview. net

What History Teaches • • In the intervening centuries some of these topics lost importance and were dropped from the mathematics curriculum. Also, the expansion of mathematics itself and of knowledge generally has compelled educators to drop other topics. Copyright 2008 www. biblicalchristianworldview. net

Isolation is Not the Answer • • But the isolation of mathematics from all applications relevant to our times cannot be tolerated, even if the inclusion of applications necessitates covering less mathematical topography. It is better to teach a little well and with understanding than a lot poorly and with no understanding. Copyright 2008 www. biblicalchristianworldview. net

Ramifications • This approach would present what is interesting, enlightening, and culturally significant, with the inclusion of only those concepts and techniques that will serve to further the liberal arts objective. Copyright 2008 www. biblicalchristianworldview. net

Specifics • • No technique for the sake of technique should be presented in the first two or three years. In other words, the material should be objective-oriented rather than subjectoriented. Copyright 2008 www. biblicalchristianworldview. net

Final Question • But what do we do for the future professional user of

Answer • Admittedly a small but appreciable percentage of the students will become mathematicians, physicists, chemists, engineers, social scientists, technicians, statisticians, actuaries, and other specialists whose work requires mathematics. Copyright 2008 www. biblicalchristianworldview. net

Answer • • Of course these students, too, should know the cultural significance of mathematics. Moreover, students who are already inclined toward a specific career will certainly take an interest in mathematics if they see how the subject is involved. Copyright 2008 www. biblicalchristianworldview. net

Recommendations • The textbooks written by Harold R. Jacobs incorporate this pedagogical thrust. – – – • Mathematics: A Human Endeavor Elementary Algebra Geometry: Seeing, Doing, Understanding The only draw back is that the life-sustaining ingredient, the Christo-centric perspective, is missing from these works. Copyright 2008 www. biblicalchristianworldview. net

Summary • • The misdirection of High School Mathematics Education: disconnection from the world of God’s making. Biblical Christian direction (or framework) for reformation of Algebra: connection to the world of God’s making. Copyright 2008 www. biblicalchristianworldview. net