What can be Christian or non-Christian about mathematics? Two plus two equals four in all classrooms, both Christian and non-Christian. Cornelius Van Til (1) refers to mathematics education as the "black beast of Christian instruction" due to its oft-misunderstood role in the Christian school. He asserts that most people, both Christian and non-Christian, see mathematics as neutral to religion. Christian parents want mathematics taught by a Christian teacher only so that a Christian atmosphere will surround their children. They do not expect that the instruction will be fundamentally any different than it would be in a public school. Rarely are there any calls to censor mathematics textbooks because seldom is there any obviously objectionable material in even the newest mathematics books - unless one disapproves of word problems where Mr. Jones bakes a cake and Mrs. Jones drives a truck. Nowhere in the textbooks is there any statement of philosophy about mathematics. Page after page has 'just the facts.'
During the last forty years of mathematics education, however, we have seen much change. In the early sixties we saw traditional mathematics instruction give way to new math. In the seventies the back-to-basics movement took over. In the eighties mathematics educators were championing the problem-solving approach. In the nineties it became standards based learning. These changes alone ought to suggest that there is more to mathematics education than 'just the facts.'
The challenge for the Christian teacher is to develop a Biblical philosophy by which to anchor his own teaching and to respond to the changes surrounding him. That is not to say that having developed such a philosophy, he will be able to teach without change. We are all sinners living in a fallen world. Just as we are continually on a path of sanctification that reforms our personal lives, so we should also be on a path of sanctification that reforms our professional lives, bringing both our curriculum and teaching methods into greater conformity with the will of God. As Van Til so humbly expressed, "Ah how large a portion of the grace of God it requires to be a teacher of the children of the covenant!"(2)
Nature and Origin of Mathematics
Let us begin our discussion of the nature of mathematics by discussing what mathematics is and what it is not. To give an admittedly simplistic definition, mathematics is the study of the numerical and spatial aspects of the universe.(3) The concept of number is used to count and the concept of space is used to measure. "How much?", "How many?", "How long?" are all questions for which mathematics has the answer.
Where did these concepts originate? When we think of God's creation, we often think only of physical objects - trees, animals, people, mountains, and oceans. Too easily, we overlook the many nonphysical aspects of these objects. Objects are hot or cold because God created temperature. Objects have a beginning and an end because God created time. Similarly, objects are countable and measurable because God created the many parts of the universe with a numerical and spatial aspect. We can count because God made objects in a discrete, countable form. We can measure because God made objects to take up measurable space.
Over the years, as man observed God's creation, he began to see more of the patterns built into it. An early observation would have been of two rocks, two trees, two horses--from which man abstracted the concept of twoness. When man selected a name for this concept he began to form the discipline of mathematics.(4) In different regions of the world different names were chosen for the concept of twoness and different symbols were chosen for writing it, but the concept was universal. The concept was of God.
Once man had given names to different numbers, the arithmetic operations followed naturally. At some point man would have observed that putting three sheep in a pen with two sheep would always make a total of five sheep, and from this he abstracted the concept of addition. Mathematics is then the discovery and organization of patterns that are built into the creation. All the arithmetic facts, all the geometric formulas, and all the algebraic properties that we teach are merely abstractions of God-created patterns in the universe.
Though all will agree that man has had an important role in the development of mathematics, the Christian and the non-Christian will disagree on the nature of that role. To the non-Christian, man's role is that of originator, or creator. John Dewey, a prominent humanist of the twentieth century, said that humans do not discover truth, they make it.(5) That philosophy is apparent in the title of a college textbook from the early seventies titled "Mathematics: the Man-Made Universe." If one starts with the assumption that there is no God, then the only reasonable conclusion is that man is the creator of mathematics. This secularist view tends to glorify man and his intellect. It considers man able, with his reason, to make sense out of the world and control his future.(6)
The Christian, however, recognizes God as the ultimate creator of all things both physical and nonphysical. God is the source of knowledge, and the mind of man is but a gift from God.(7) Consequently, the Christian defines man's role as that of discoverer and developer of mathematics, rather than of ultimate creator. Indeed, since mathematics has been an important tool for understanding and thereby ruling over the earth, the development of mathematics can be viewed as an act of man's obedience to the cultural mandate. Be fruitful and increase in number; fill the earth and subdue it. Rule over the fish of the sea and the birds of the air and over every living creature that moves on the ground. (Genesis 1:28)
And after all of the labor of discovering and writing out principles of mathematics, how do we know that these same principles will still hold true tomorrow? The non-Christian has no answer for such a question. He can only hope for the best. The Christian knows that the same God who created the universe sustains it day in and day out. Harro W. Van Brummelen summed it up well in his statement, "Laws of mathematics are dependable not because man with his logical reasoning created a fool proof system, but because God in His constant faithfulness embedded these unchangeable laws in His creation."(8)
In order to use mathematics properly we must also understand its limitations. Mathematics is a tool for describing the creation, not the creator. Every attempt to define a mathematical model for the triune God fails in some respect. When we count each personage of the Godhead as a discrete entity, we have three gods. When we illustrate God as a whole divided into three parts, each personage becomes only one-third god. These are both very unsatisfactory models! This should not surprise the thoughtful Christian, as the creator is never bound by his creation. Can you fathom the mysteries of God? Can you probe the limits of the Almighty? (Job 11:7)
Even within the creation, mathematics is only one aspect of a multi-faceted universe. It does help us describe such aspects as temperature, position, and time. It does not help us describe such aspects as emotion, beauty, and commitment. We can never reduce all of creation to a mathematical model. Interestingly enough there have been attempts to do just that, which we will discuss in the next section.
Secular Influences on Twentieth Century American Mathematics Education
Two non-Christian philosophies that have influenced mathematics education in the United States in the second half of the twentieth century are Logicism and Formalism. Though their influence in the present day curriculum is waning, they are of particular importance to me because they were most prominent during the sixties when I was in high school and college. My education from Kindergarten through graduate school was entirely from secular institutions. Hence, my own personal training in mathematics was permeated by these philosophies and it carried over heavily to my early years of teaching. Only by studying these philosophies in the light of scripture have I been able to move my own teaching along that path of sanctification.
Logicism is the mathematical outworking of rationalism, which traces its roots back to the Enlightenment of the eighteenth century. It holds that reason, or analytic thought, is the source of all truth.(9) The logicist begins with a small number of truths that are recognized to be unverifiable, but are accepted on faith. From these assumptions is logically deduced all of mathematics and all of accepted reality.(10) Thus, human reason, rather than Biblical revelation, becomes the source of all knowledge.
In mathematics, these unverifiable truths originally centered around real numbers. In the late 19th century through the work of Giuseppi Peano and Georg Cantor the foundations of mathematics were pushed down to the more basic system of natural numbers. In the early 20th century Ernst Zermelo put forth a set of axioms, which has come to be called set theory, for describing the system of natural numbers in terms of concepts of sets. Bertrand Russell and Alfred Whitehead took it one step further in their Principia Mathematica by trying to show that all of mathematics can be made to rest on propositions of logic. Ultimately, Logicism is an inherently flawed system. The work of Kurt Godel in 1931 demonstrated that a logical system could never be both complete and consistent, and hence Logicism could never adequately serve as a philosophical foundation for all of mathematics, much less for all of reality.(11)
Formalism is the mathematical outworking of Postmodernism. The postmodernist does not believe there is any final truth.(12) Thus the formalist begins with basic assumptions that are set forth by man, regardless of whether they have any relationship to reality. He then logically deduces a system of statements that are never true or false, only valid or invalid based upon the original assumptions. He has no interest in trying to apply these results to the real world because he considers reality unknowable.(13)
Formalism originated in the writings of David Hilbert who asserted that the basic objects of mathematical thought are the mathematical symbols themselves. The mathematician should not, he believed, be concerned with attaching any meaning to those symbols. A strong adherent to this philosophy was the mathematician Godfrey Hardy who believed that a mathematical topic was only to be considered pure mathematics if it was utterly useless. He even boasted that: "I have never done anything 'useful.' No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world." The false nature of this philosophy has been exposed by the work of other mathematicians who took Hardy's work and demonstrated its roots in God's creation. Hardy's Law became central in the understanding of Rh-blood groups and in the treatment of hemolytic disease in newborns. In addition, his investigations of Riemann's Zeta function have formed a foundation for the theory of pyrometry (temperatures of furnaces).(14)
Throughout history God has used both Christians and non-Christians to unfold the mysteries of His creation. Adherents of both Logicism and Formalism have had positive effects on the development of mathematics. The emergence of set theory brought unification to many diverse branches of mathematics and introduced elegant notation for expressing certain concepts.(15) Due to the influence of Formalism on mathematics education, more emphasis was placed on the understanding of the foundations of the number system, its algorithms and properties. For more able students this produced a broader understanding of the nature of mathematics and provided a better foundation for more advanced study.
A negative consequence was that in some textbooks, set theory became the basis for introducing mathematical concepts at the lowest grade levels. Instead of presenting numbers and operations as abstractions of patterns seen in God's creation, a number was taught to be a name of a class of sets and addition was introduced as the union of disjoint sets. Throughout the grades, application of mathematics to the real world was de-emphasized. The number of pictures and word problems diminished in the textbooks. Sets, rather than God's created world, became the motivational source of mathematics. This level of abstractness especially harmed less able students. Many failed to learn basic skills or to see any relevance in the study of mathematics.
For the past ten years the most dominant influence on mathematics education in the United Sates has been the Principles and Standards for School Mathematics published by the National Council of Teachers of Mathematics first in 1988 and revised in 2000. There is much good to be said for these standards. They have championed the need to have mathematical understandings grow out of a student's experiences in the real world. They have discarded the straightjacket of formalism for an approach that seeks to engage the student in meaningful problem solving at every level. They have also called for a change in the often-shabby approach to educating the less able students in our schools.
However, one does not have to read far to experience the unbiblical philosophies of humanism and rationalism that pervade the standards. We are boldly told that: 1. reasoning is the standard of truth in mathematics,(16) and 2. a major goal of school mathematics programs is to create autonomous learners. By autonomous, educators mean students who will construct their own ideas about mathematics and decide their own truth without having to be told.(17) In a special report to the nation prepared by the National Research Council in 1989 we find these supporting statements: "Mathematics does provide one of the few disciplines in which the growing student can, by exercising only the power inherent in his or her own mind, reach conclusions with full assurance. More than most other school subjects, mathematics offers special opportunities for children to learn the power of thought as distinct from the power of authority."(18)
Broad Objectives of Christian Mathematics Education
As a young teacher in a Christian school, I struggled with the question, "How do I put God into my mathematics class?" I listened to many lectures on the integration of the Bible and learning, but they did little to help. I always felt that I was trying to impose something artificial on mathematics. Yes, there were numbers in the Bible, but pointing that out hardly made my teaching distinctively Christian. I finally began to make progress along that path of sanctification in my profession when I came to realize that I did not have to put God into mathematics. He is already there! My role is to reveal Him to my students. I should be helping students to remove the blinders that sin puts on their eyes so that they can see the character of God revealed in every part of the creation.(19) For in Him we live and move and have our being. (Acts 17:28)
Furthermore, the content in my classroom may well be the same as the content in a secular school not because I am failing to teach Christianly, but because that content is rooted in God's creation wherein all Christians and non-Christians function. The difference is rather in what I do with that mathematical content. Students need to experience mathematical content in the context of its true meaning. They need to be shown the proper place of mathematics in God's creation. They need to understand that it is only as we combine human reasoning with a faith commitment to the God of the Bible that we will come to know truth. And God's truth is not passive; it calls people to respond to Him in love and service. Hence, for me to teach with meaning requires that I give direction to the development of students' lives.(20)
Teaching mathematics with meaning is summarized in the following three broad goals.
The study of mathematics should reveal the God of the heavens and earth by whose Word all things were brought into being and are sustained moment by moment.(21)
Everything God created reveals His nature. For since the creation of the world God's invisible qualities--His eternal power and divine nature--have been clearly seen, being understood from what has been made, so that men are without excuse. (Romans 1:20) However, when man fell into sin, he lost the ability to see God plainly in the creation. In God's love, He gave man the scriptures to more clearly reveal who He is and what His claim is on us. Finally, in the ultimate act of love, God gave His only son to restore the bond between God and man that was broken by sin. Once the student comes to know God through Christ, his eyes will be opened to see the truth of God in creation. As the student explores the depths and details of that creation in the classroom, he will be adding depth and detail to his understanding of God.
The study of mathematics should call the student to a life of service, using his God-given gifts to glorify his creator and help his fellow man.(22)
God's truth is distinguished from the world's idea of truth in that it is not neutral or value-free. Neutral facts make no claim on the student. Indeed the autonomous student is free to not only construct his own truth, but to decide to act or not act on that truth. God's truth is never neutral. God's truth is centered in the person of Christ. I am the Way and the Truth and the Life. (John 14:6) God's truth calls for a response. And He died for all, that those who live should no longer live for themselves but for Him who died for them and was raised again. (II Cor 5:19) God has given man responsibilities in this creation, both as keepers of the creation and as reconcilers of this fallen world to Him. As the student grows in his knowledge of the truth of God through his study of the creation, he will grow in his commitment to serve Him in the everyday activities of life. We know that we have come to know Him if we obey His commands. (I John 2:3)
The study of mathematics should lead the student into a deeper unity with the Lord that has its outworkings in increased love, praise, and worship of God.(23)
Ascribe to the Lord the glory due his Name; worship the Lord in the splendor of his holiness…The voice of the Lord thunders over the mighty waters…The voice of the Lord breaks the cedars… The voice of the Lord strikes with flashes of lightning…The voice of the Lord shakes the desert…And in his temple all cry, "Glory!" Psalm 29:2-9
In creation we see a God who out of love formed a world marvelously attuned to our needs. He placed the earth at just the right distance from the sun. He surrounded it with the perfect atmospheric conditions needed by our bodies. He daily provides the sunlight and rain to grow our food. He gave us a mind with which to investigate that world. Indeed, He is a loving God, and His love calls forth a response in us of love towards Him. We love because He first loved us. (I John 4:19) When the student comes to see that the earth really is the Lord's, his response cannot help but be to love Him and give Him the glory due His name.
Implications for the Curriculum
The implications of these goals for the mathematics curriculum in a Christian school are far-reaching. In some of these areas we will find ourselves in concert with the best of secular education around us, because all truth is God's truth and common grace abounds. In other areas we will be distinctively different.
Mathematical principles must be presented as generalizations of man's experiences in the creation, not as brilliant creations of the human mind.
In the study of geometry, it was traditional to present each new theorem as a logical deduction from previous theorems or postulates. Such teaching gave the impression that someone in days past sat down and developed the whole of geometry by placing one logical precept upon another. Nothing could be farther from the truth. Most theorems in geometry originated as the result of inductive reasoning based on experiences of everyday life in the creation and came to be proven deductively only well after their general acceptance.
Teaching in the classroom should mirror this reality. A student should normally be given time to explore a topic intuitively before it is handed to him in polished form. This will give him the opportunity to discover and organize mathematical patterns for himself. Such opportunities will both improve his problem-solving skills and help him to appreciate how mathematics has been developed in the past and will continue to be developed in the future.
The proper role of reasoning should be discussed. It is not, as the Logicist would claim, "an autonomous guide capable by itself of leading us to truth. Our reason always follows our heart."(24) Our heart is either committed to the true God of Scripture or to some idol. All the theorems of mathematics rest ultimately on assumptions that remain unverifiable except through faith.(25) The Christian student must be taught to use his reasoning in concert with God's revealed Word to find truth.(26)
Mathematics must be taught in a way that reflects the unity of all things in the creation.(27)
Christ is the light in which we see and understand everything around us. The Christian school must teach all subjects as parts of an integrated whole that is bound together at its core by the Word of God. In the mathematics classroom, we must show that every part of creation has a numerical and spatial aspect. Classroom activities and projects should illustrate the use of mathematics in fields of economics, science, history, politics, art, music, language, health, athletics, business, etc. Care should be taken to distinguish non-Christian views that man or even mathematics is the source of unity in these diverse fields from the Christian view that it is a reflection of God's unity. They exchanged the truth of God for a lie, and worshiped and served created things rather than the Creator--who is forever praised. Amen. (Romans 1:25)
At the same time, the student must be shown that mathematics itself has many different aspects. There is a language of mathematics that must be mastered by the student so that he can accurately share his mathematical understandings with others. There is an aesthetic aspect to mathematics that can be seen in the elegant solution to a difficult problem.
There is a history to mathematics that enables us to see how man has lived out God's command to have dominion over the earth. Students should be shown how in the historical development of mathematics the level of complexity of mathematics was related to the needs of the society. They should be exposed to the variety of philosophies that have emerged through the ages. They should be challenged to look for ways in which mathematics has influenced our modern culture and how our culture has influenced the development of mathematics.(28)
Mathematics education must be a dynamic discipline.
As our culture changes, the topics covered in the classroom must be reevaluated and modified to ensure that the student will be prepared to serve God in today's world.(29) Mathematics must never be taught as an end in and of itself. Christian teachers are in danger of doing just that when they cling to favorite topics of the past and refuse to recognize that God's world is not static. The use of calculators and computers has fundamentally changed what mathematics people need to know for everyday life and how new mathematical ideas are discovered. New educational research, when it is demonstrated to be valid, should help the Christian teacher to better understand how students learn. The Christian teacher's own desire to know God more fully should keep him exploring more deeply into the creation in ways that continually renew his teaching.
Mathematics and values must be taught together.
It is not possible to separate religious values from true education. Learning to do statistics, for example, gives the student a tool for functioning in the creation. But how is he to use that tool? If we do not teach that, we have not truly educated the student. Used properly, statistics can convey complex data in easy to understand visual formats that enable others to make informed, intelligent decisions. A well-known book among mathematicians is titled "How to Lie with Statistics" by Darrell Huff and Irving Geis. It exposes the common methods for using statistics to mislead the public. When we teach mathematics as value free, we open the door for just such misuse. Indeed our present day advertisement industry has raised the skill of lying with statistics to the level of high art. Godly education instructs the student in how to use his skills to honor God and to expose the abuses of others. It produces a changed heart and an altered life, growing in the fear of God.(30)
Learning mathematics will require hard work on the part of the teacher and the student.
We live in a fallen world and sin rears its ugly head at every turn. Our attitudes towards work are no exception. Children have a God-given curiosity, but that is not the same as a love of learning. Adults are increasingly pleasure oriented rather that work oriented. Hence, we often see a decline in a student's enthusiasm with school as he progresses up through the grades, a decline that is frequently caused by the sin of laziness on the part of the teacher or student, or both. As Christian teachers, we seek to give students not only the tools to learn about God's creation, but the desire to use those tools. Love is a powerful motivator. We love God and our fellow man because God first loved us. (I John 4:7,19) When our teaching is permeated with our love for mathematics and the God who created it, we inspire our students to delve more deeply into that which gives us so much joy. Love requires sacrifice. God so loved us that He sent His only son to die for us. (John 3:16) Teaching with love and learning with love will require both the teacher and the student to give of themselves wholeheartedly, sacrificing worldly pleasures to His purposes.(31)
Mathematics must be taught by a Christian teacher who knows God and knows himself.
God's Word teaches that we cannot begin to build a new life in Christ until we have first broken down the old life of sin. That is true for both the student and the teacher. The Christian teacher is one whose eyes have been opened to see the depths of his own depravity and has acknowledged his complete unworthiness to merit God's grace. He has been redeemed by the substitutionary death of Jesus Christ and brought to faith and repentance through the renewing work of the Holy Spirit. He is daily kept in faith by the almighty power of God and enjoys the security of knowing that nothing can separate Him from the love of Christ.(32)
The Christian teacher understands that the Word of God has spoken all of creation into being. He recognizes that his ability to acquire reliable knowledge about that creation is based on the faithfulness with which God keeps His covenant promises and maintains His creation.(33) He understands that by studying the creation he can learn to know God more fully and be brought into even deeper communion with Him. He views work and the study of the creation as a form of praise and worship of God.
The Christian teacher knows that his worldview, Godly or otherwise, will gradually condition the worldview of his students.(34) He does not fool himself into thinking that his teaching or his life witness will bring his students to Christ. He knows that salvation is the province of God alone. Nevertheless, he expects that God can and will use him to prepare the way for the working of the Holy Spirit in the lives of his students. In the words of Cornelius Van Til, "The Christian teacher knows himself, knows the subject, and knows the child. He has the full assurance of the absolute fruitfulness of his work. He labors in the dawn of everlasting results."(35)
Glorifying God in My Classroom
Anyone who has spent significant time in Christian education has found himself trying to explain and defend the role of the Christian school to those who do not see the point of its existence. To some of our critics we should teach Bible non-stop because knowing Christ is all there is. To others of our critics we should confine the teaching of religion to our churches and send our children to the local public school to learn an objective view of the world. In "Recovering the Lost Tools of Learning," Douglas Wilson presents a charming analogy to explain the role of Christian education using the sun to represent Christ as the light of the world. (John 8:12) Wilson likens the first criticism to a person who ignores the world and spends all day staring at the sun. He likens the second criticism to a person who blocks the sun and attempts to study the world in the darkness that results. By contrast, the student in a Christian school studies the world in the light of the sun. In [Christ] are hidden all the treasures of wisdom and knowledge. (Col 2:3)
In this last part of the paper, I will describe how I attempt to bring Godly meaning to the study of mathematics at Delaware County Christian School, using the light of Christ. I am encouraged by the realization that sanctification is a lifelong process. What follows is a description of where I perceive that I am on the path of sanctification in my profession.
Mathematical principles must be presented as generalizations of man's experiences in the creation, not as brilliant creations of the human mind.
For students to see the reflection of God in mathematics, they have to see that mathematics is rooted in God's creation. In geometry, we discuss the fact that lines and circles and polygons are simply abstractions of the shapes that God used in building His creation. As a result of my reading for this paper, I have come to realize that I must be more careful in such discussions not to hold up the mathematical model as the ideal and the God-created object as an imperfect representation. Rather, it is the mathematical abstraction that imperfectly attempts to describe the great complexity of God's world.
Geometry also provides an ideal backdrop for discussing the role of reasoning in mathematics as well as in doctrine. Inductive reasoning consists of drawing a general conclusion based on the observation of a pattern. It is perhaps the primary tool God has given the mathematician for discovering patterns in His creation. In the computer laboratory, my students have the opportunity to discover geometric patterns for themselves. By constructing a variety of isosceles triangles on the computer and measuring their angles, for example, they can observe the pattern that the base angles of such triangles are congruent. However, it is a leap of faith to say that something observed in three or four examples is necessarily true for all such items. Hence, we then proceed to apply deductive reasoning to support such a hypothesis.
Deductive reasoning consists of drawing a general conclusion by logical inference using only accepted facts. Supporting our observation from the laboratory by writing a deductive proof then strengthens our confidence in the truthfulness of our observation. However, the important realization for the Christian student is that he is still making a leap of faith. Our definition of deductive reasoning above refers to "accepted facts." How are we to establish these "facts", since they cannot be proven? They are, in fact, faith agreements. In geometry, one of these accepted facts is that "two points determine a line." This "fact" is unprovable. Nevertheless, it is such an obvious statement, that our inability to prove it takes nothing away from our certainty that it and anything we deduce properly from it are true.
We then discuss how this relates to the way we reason in doctrinal matters. "The Bible says that Christ died to save sinners. I am a sinner. Hence Christ died for me." That is deductive reasoning. But its validity is rooted in the faith agreement that there is a God and that the Bible is truly His Word. We cannot logically prove that God exists anymore than we can logically prove that two points determine a line. Yet with the eyes of faith both are fully believed. Through a discussion of these issues in the classroom, I want my students to understand that reasoning by itself cannot bring us to an understanding of truth in mathematics or in doctrine. Truth comes only when reasoning is rightly applied by faith to that which God reveals to us in His Word and in His creation.
Mathematics must be taught in a way that reflects the unity of all things in the creation.
To demonstrate the unity of God's creation, I bring to my courses problems that illustrate the role of mathematics in the investigation and understanding of many different fields of study in the creation. In calculus we just recently finished an assignment where students analyzed data about the distribution of income in the United States over the past 70 years. They used methods of calculus to quantify the degree of income inequity year by year and computer generated graphs to visually display the patterns of change. Earlier in the year, the class did a computer laboratory activity with skydiving. (They begged me to make it a field trip!). Using computer generated graphs to model the fall of the skydiver, we were able to analyze the perfect time to pull the rip cord so as to enjoy the most free-fall while preserving just enough time to float gently to a safe landing. Throughout the course, I try to bring in non-standard examples where calculus helps us to understand or predict human or physical phenomena. We model traffic flow on a freeway. We determine the frequency of drug doses needed to maintain a certain safe level of a drug in the body. We calculate the number of years needed to remove pollutants naturally from a body of water based on its rate of outflow. With the eyes of faith, each example becomes one more piece of evidence of the divine unity behind the creation and one more glimpse into the depths of God's genius.
In geometry students research a variety of mathematical patterns to see how they are evidenced in the creation. They study spirals and learn how they describe the shape God gave to the chambered nautilus and the horns of the barbery sheep. They learn about the Fibonacci sequence and how it describes the arrangement of petals on a flower or the ancestors of a honeybee. They investigate the elliptical paths God gave to the planets and comets as they orbit the sun.
Equally important is to help students see how God gave many different aspects to mathematics and each is important in understanding, using, and appreciating God's gift of mathematics fully. Of utmost importance is God's gift of language. God created mankind to live and work in community with each other and language is one of the tools of community living. I require students to learn the vocabulary of mathematics so that they can read and comprehend mathematical explanations written by others. Especially in algebra, I stress learning to use mathematical symbolism properly so that students can communicate their mathematical ideas with others. God's gift of language also gives clarity and organization to the student's own thinking. Thus by teaching students the language of mathematics, I am laying a foundation for the learning of more complex mathematical ideas in the future.
Because God created time, mathematics has a history. When I teach trigonometry, I discuss how God enabled the ancient Babylonians to develop a working knowledge of geometry and trigonometry to help them control the floodwaters of the Tigris and Euphrates Rivers. In calculus, we discuss the historical significance of the discovery of the Fundamental Theorem of Calculus and some of the all-too-human jealousies among mathematicians that harmed the development of mathematics. In algebra and calculus, we talk about how the rise of computers and calculators is actively changing the nature of mathematics in our own society. By presenting mathematics in its historical context, I try to help my students see that mathematics results from man's obedience to the cultural mandate and from God's goodness in opening up to him more of the mysteries of His creation.
Because God is beautiful, we see that He has imbedded great beauty in mathematics. In the classroom we see the marvelous way in which different branches of mathematics are intertwined and how exciting it is to see a complex idea made simple by the application of the proper theorem. Though not every student sees the same beauty in mathematics, I take great delight in seeing the acknowledging smiles on even a few faces when our class discussion brings us to a moment of great clarity on a mathematical idea.
Not only is there an aesthetic aspect to mathematics, but we also see a mathematical aspect to aesthetics, or fine arts. Teaching geometry gives a wonderful opportunity to use mathematics to create art. After discussing the mathematical properties of tessellations, students get to create their own tessellations, first by using colored shapes on the computer and later by cutting them out of construction paper. We also study the mathematics behind drawing in perspective, and learn how to use the computer to facilitate such drawings. Each year several students in geometry research the golden ratio and present to the class the influence this mathematical concept had on classical art and architecture.
Mathematics education must be a dynamic discipline.
God is alive and at work in His world. New mathematics is constantly being developed, and the widespread use of computers enables old mathematics to be applied in new ways. I have added an introduction to fractals in my geometry course so that students will better understand the developing nature of mathematics. We learn how fractal ideas are used in understanding the measure of a coastline and in describing the dimension of new kinds of curves. In calculus we use direction fields and Euler's Method to work with differential equations that would be unsolvable by more traditional methods. Assisted by calculators and computers, Euler's Method enables us explore problems with stable and unstable equilibriums, and to carry out the skydiving activity discussed earlier.
As we enter the twenty-first century, we find that employers now seek different mathematical skills than in years past. To respond to these changes I have been continuing to shift the emphasis in my algebra and calculus classes away from the predominance of analytic skills to a broader based approach that involves teaching concepts graphically, numerically, and analytically. For example, in previous years the primary means of solving an equation was to follow a pencil and paper algorithm. Though I still teach such algorithms, we spend less time perfecting these skills on complex equations, because computers and even calculators are capable of performing these computations quickly and accurately. Instead, we spend more time learning how to use graphs to solve equations, how to evaluate the reasonableness of a proposed solution, and how to interpret the solutions. In addition I give students more opportunities to discuss their ideas with a classmate and to work cooperatively towards the solution to a problem. These are all skills that are important in the way mathematics is being used in God's creation today.
Mathematics and values must be taught together.
Douglas Wilson uses the phrase that a sinner educated apart from the Bible simply creates a cleverer sinner.(36) As mathematics is taught in a Christian school, it needs to be coupled with direction for using that knowledge in God honoring ways. As we study percents in algebra and their application to calculating investments, we discuss wise investment principles and good stewardship of the money God gives us. In completing the calculus project discussed earlier on income inequity in the United states, my students are asked to document the historical events and government policies that may have contributed positively or negatively to the level of inequity. In this way they can begin to see how mathematical knowledge can be used to guide the formation of government policy and bring about positive changes in our society.
The project my students do in geometry opens the opportunity to discuss academic integrity so that they learn to give proper credit for the use of information they have gotten from others. Everyday activities of turning in homework and taking tests give opportunities to discuss personal honesty. In the enforcement of school rules, I have the opportunity to teach students the difference between a begrudging, half-hearted obedience to authority and the humble, wholehearted obedience that the Bible teaches.
However, the strongest voice for teaching values is not the classroom discussions or even the individual counseling, but the silent witness that my own life presents. It is a challenge that I take very seriously even as I admit my failings. To teach students obedience to authority, I must model the same in my responses to administrative authority over me. To teach students to treat each other with kindness and forgiveness, I must respond to them with love and forbearance. To teach students responsibility in their daily work, I must demonstrate responsibility in my teaching--being prepared for class every day, grading student papers promptly, and carrying out administrative instructions fully and on time. To teach students honesty, I must be honest before my students, admitting my mistakes and taking responsibility for my failures. The most powerful teaching is accomplished when teacher and student travel together down the road of discipleship.(37)
Learning mathematics will require hard work on the part of the teacher and the student.
I love teaching and doing mathematics. It is my passion. It is also my living sacrifice to the Lord. To do it well requires that I spend much time outside of school learning more about mathematics and developing better ways to teach it to my students. I will summarize here a few of my most recent advances in mathematics. A few years ago I started learning about fractals and how to interweave them in the curriculum. I have used them to advance students' understanding of complex numbers and to experience some of the thrill of discovering new mathematical ideas for themselves. Two summers ago I attended a conference where we explored ways to teach students to tackle more real-world problems in mathematics, problems in which students have to wrestle with social and economic values in their solution of the problem. I have been using some of these examples to enrich the curriculum in our Mathematics Club. Last summer I learned to use the Geometer's Sketchpad program as a tool to dynamically illustrate concepts of calculus. My students responded very enthusiastically this year to lessons where I used these demonstrations to teach new concepts. This past Christmas vacation I had my son show me how to construct a web page. I worked for the following two months to create a web page that now serves as a homework resource for my students and a communication link with their parents.
Through my love for God and His creation, particularly the mathematics of that creation, I desire to inspire in my students a similar love for the Lord and the world in which He has placed us. That love will call for a response of sacrifice from them. I expect them to invest themselves wholly in the task of learning. Whatever you do, work at it with all your heart, as working for the Lord, not for men. (Col 3:23) Students cannot learn mathematics by sitting idly in the classroom, however quiet and well mannered their behavior. Hence I prod them with questions, solicit their ideas, and encourage them to challenge results that do not make sense to them. Students cannot learn mathematics when they neglect homework. Consequently, I check their homework daily and use every means at my disposal to 'force' them to be diligent. They also cannot learn mathematics when they approach it as a mindless task of repeating processes demonstrated in the classroom. This is a much more difficult problem to deal with because educators themselves are often at the root of this problem. Too frequently, teachers reward students with high grades when they can successfully perform mathematical algorithms, but are unable to apply them to the meaningful solution of real-world problems. This is unfortunate. Mathematics is a tool for doing God's work in the creation. If the student cannot properly apply mathematical processes to meaningful problems, he has no true knowledge. I am constantly confronting students with problems that require them to think about the meaning and application of what they are doing. I also work to develop tests that measure student's ability to use mathematics in meaningful ways.
Mathematics must be taught by a Christian teacher who knows God and knows himself.
The Lord took me, a self-centered sinner, and for His glory changed my heart to seek His will. He taught me through the words of scripture that Jesus died for me, to pay the penalty for my sin. (Eph. 1:3-12) His Spirit brought renewal to my being. He gave me purpose and direction in life. He holds me in His loving arms and assures me of spending eternity in His presence.
He called me to a life of service as a mathematics teacher. I am encouraged by the promise that His strength is made perfect in my weakness. (II Cor. 12:9) Being a Christian teacher requires that I remain close to the Lord. The Lord can only work through me as I allow His standards to be my standards. Be holy because I, the Lord your God, am holy. (Lev. 19:2)
In the classroom I strive, however imperfectly, to model the example of Christ as the great teacher. He used a variety of teaching methods to reach the audience at hand. He made time for every person who sought knowledge from Him. He held up the highest of standards. He showed compassion when appropriate and judgment when necessary. His love was always evident.
Most of all I seek to teach that Christ is the truth in everything. It is He to whom all of creation points. Numbers and space exist because God spoke them into being for His purposes. His power upholds them, moment by moment. All of mathematics is but one of the wonderful gifts of God's love.(38) It is His truth and His creation that we study. It is His name that we praise for all that exists. And it is for His glory that we live.
For from Him and through Him and to Him are all things. To Him be the glory forever! Amen. Romans 11:36
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