Dr. Harro Van Brummelen,
dean of the School of Education at Trinity Western University in British Columbia, Canada, taught mathematics at various grade levels for 14 years. This article is based on the section on mathematics in the second edition of his book "Steppingstones to Curriculum," published by ACSI. Dr. Van Brummelen has lectured worldwide on Christian education and has taught ACSI-sponsored courses for teachers in central and eastern Europe.
In its discussion of standard notation of numbers, Charis Mathematics asks students to work with numbers that represent very large sizes and distances in the solar system, the Milky Way, and the universe. It also asks them to look at numbers describing very small things—cells, viruses, molecules, and atoms. It quotes physicist Paul Davies that the universe is “full of stunning surprises” and that “it is difficult not to be struck by some surprisingly fortuitous accidents without which our existence would be impossible.” It then asks students to consider, as part of their work on extreme numbers, whether our presence on the earth could be a cosmic accident or whether there is design behind the universe. The authors want students not only to learn standard notation but also to gain a sense of wonder about the universe and to reflect on its origins and their place in it (Shortt 1996).
This example shows that mathematics is more than a construction of the mind—and that as teachers we must help our students see the finger and the glory of God as they study mathematics. Mathematics is not just a series of formal, human constructions that we manipulate using rules of logic. That approach, in fact, implicitly gives students three ideas: first, that as humans we control the universe; second, that we can solve its problems through human reason; and third, that we can make progress as a society without taking into account our religious and ethical groundings. At best, students are left with a dualistic view of life, concluding that faith and values may be important in their personal lives, but not as they try to have an impact on society.
How Did Mathematics Originate?
Actually, mathematics originated not in our minds but from human experience and activity with two aspects of God’s created reality: numbers and space. It takes faith in the continuing validity of mathematical laws to do and use mathematics. Thus the study of mathematics brings out a sense of awe and wonder at the design and order of God’s creation as well as at the faithfulness, majesty, and transcendence of God. Once people discovered mathematical results through observation and informal reasoning, they developed symbols to represent their conclusions. Those symbols stand for real and changeless created entities.
Only after using the results for many years or even centuries did mathematicians prove their findings logically. They based their proofs on “self-evident” assumptions that, it turned out, did not always hold. Rejecting a postulate that held only for geometry on a flat surface led, for instance, to non-Euclidean geometry. Moreover, in 1931 the German mathematician Gödel proved that it is impossible to design a mathematical system that is both complete and consistent. That means that mathematicians must live with some baffling paradoxes. They can do so because they have faith in the basic order and law structure that God embedded in the universe.
Number and Space in Mathematics
The first key meaning in mathematics is discrete numbers. Numbers function according to God-given laws that we discover from experience. People have named numbers and have gradually improved their designation. Our present system is probably too firmly entrenched to make further advancements, even though some argue that changing from a base 10 to a base 12 number system would make our mathematical lives easier. However, even when human descriptions of the underlying structures change, basic mathematical laws stay constant.
The second key meaning in mathematics is continuous space. This concept cannot be reduced to or explained in terms of discrete numbers. We have no way to designate all numbers on the number line as fractions or as terminating or repeating decimals, even though there are infinite amounts of each of these. Yet we need numbers to describe space and figures in space: a plane has two dimensions; a quadrilateral, four sides. In other words, we use numbers to think about geometry, but numbers or numerical descriptions will never do full justice to geometry. Mathematics may, however, develop understanding of the numerical and spatial aspects at the same time, as in coordinate geometry.
The Role of Mathematics in Culture
Mathematics plays an essential but limited role in other knowledge. It points beyond itself to the other aspects of reality. Mathematical models are useful tools in physics, psychology, and economics, for instance, but they never fully describe the meaning of a situation. Physicists, psychologists, and politicians must consider other aspects of reality than the mathematical ones to analyze situations and make responsible decisions. Note also that mathematics cannot be equated with logic or analytical reasoning. The content of mathematics involves the two simplest aspects of reality. Therefore we can prove mathematical results using logic more clearly and more convincingly than results in other areas of knowledge—but the basic principles of logic apply there just as much.
Why Teach and Learn Mathematics?
So why do we teach mathematics to our students? First, we want to deepen students’ understanding of God’s creation and of how mathematics helps them fulfill their calling. Students can abstract the mathematical aspects of real-life situations, analyze them, and use the results of their learning in applications. They can explore how we use math in science (e.g., leaf surface area and water loss). However, applications go beyond science. Mathematics is part and parcel of human culture.
We teach mathematics so that our students will:
- Recognize that God is faithful and reliable in upholding the world through orderly mathematical patterns, laws, and structures that He embedded in His creation.
- Gain understandings of the concepts of number and space and their interrelationships.
- Deepen awareness of mathematics as a functional tool in solving everyday problems in diverse settings.
- Experience mathematics as a developing science. Mathematics is not a fixed body of knowledge but grows as cultures develop—and is fallible. For instance, the Babylonians for years used an incorrect formula for the area of quadrilateral fields, using the results to levy taxes.
How Do We Implement These Goals?
These goals suggest that we should base classroom mathematics on everyday situations. We give students time to explore situations, pursue hunches, and draw conclusions from their observations. Such situations may involve experiences from students’ lives outside school or classroom explorations with manipulatives, science activities, human geography, or business settings.
We also give students more formal instruction and practice to ensure they gain a clear grasp of math concepts and how to apply them. It is helpful to relate such direct instruction to situations in which students have tried to solve problems using a variety of strategies.
With the use of calculators, the scope and nature of numerical practice has changed, but students still need dexterity with basic number skills and estimation. Mental mathematics still has a place in grades two to six, and some teachers report that it can add elements of accomplishment, fun, and joy to learning. Besides, students need drill to reinforce concepts and algorithms.
Students should experience the use of mathematics in various relevant contexts. They should apply math results from one situation to a variety of settings. The broad scope of math applications lends itself to meaningful problem solving. Mathematics is not an isolated, self-sufficient body of knowledge, but an indispensable tool in most areas of life.
Textbook problems tend to ask students to apply only one math algorithm at a time. This promotes convergent thinking. Students look for one pattern and apply it mechanically. Real problem-solving ability depends on critical and creative thinking. That is why the National Council of Teachers of Mathematics in its 2000 standards document promotes both conceptual understanding and procedural competence, with an emphasis on conjecturing, inventing, reasoning, and solving mathematical problems in various settings. This goal requires active involvement in transforming and simplifying given problems, writing different versions, modifying problems to yield different solutions, and constructing original problems. Students need this type of experience with everyday graphical, statistical, and financial data, using calculators and computers when appropriate.
Mathematics in a Meaningful Context
Leaders in society today often base economic and political decisions mainly on the numerical aspects of a situation. They use polls and look at the short-term balance sheet. Students should learn that we cannot reduce complex situations and decisions to mathematics alone, even when quantitative aspects provide useful insights. Even a decision to buy skis should involve deeper questions than how much they cost. What should be our financial priorities? Why? If we neglect those matters in math class, students will get a skewed view of life’s priorities. Yet mathematics texts seldom venture beyond calculating what students can buy in the marketplace with the money they have, in effect promoting materialism and consumerism.
The history of mathematics should also be part of the curriculum. Students should see how mathematicians unfold new concepts and techniques, often in informal, intuitive, and creative ways. They learn that mathematics is an important but limited part of culture, and that intuition and value judgments affect its development. Historical approaches can make clear how mathematical problem posing and solving has contributed to cultural development. At the high school level, we can also show how different worldviews have led to different approaches to mathematics. For instance, the Greeks accepted a certain type of mathematical reasoning as foundational to life (and this still influences us today!), while the Chinese emphasis on harmony, balance, and intuitive pedagogy led mathematics in a different direction—but an equally powerful one (see Chapter 2 in Howell and Bradley).
Mathematics Is Not Neutral
Teaching and learning mathematics is not a neutral activity. Textbook examples and problems often promote an individualistic, materialistic way of life. And current mathematics learning often values individual, reproductive, and formal approaches rather than the exploratory and applied work involved in such activities as cooperating in projects, discussing problems, studying the origins of numbers and geometry, pursuing open-ended investigations, and exploring value-rich issues such as misleading uses of statistics (Ernest 1991).
In short, to support our aims, we may need to design alternate examples and learning strategies. The Christian Charis Mathematics Project (Shortt 1996, 1997) includes examples like the following:
- How much is your gift worth? (giving to churches and charities in relation to one’s means: calculations with percentages)
- Fractals (the complexity of the universe: ratios, perimeters, and accurate constructions)
- The moment of truth (respect for reasoning and truth based on the history of prime numbers)
- But can you afford it? (personal and ethical responsibility when making financial decisions; calculating interest)
Let’s make sure that in our schools mathematics also contributes to the recognition of the greatness of God and what it means to serve Him in obedience. Mathematics can be exciting—and can help our students apply their learning in meaningful ways that will help them to work and take care of God’s earth (Genesis 2:15).
References
Ernest, P. 1991. The philosophy of mathematics education. London: Falmer.
Howell, Russell W. and W. James Bradley, eds. 2001. Mathematics in a postmodern age: A Christian perspective. Grand Rapids: Eerdmans.
Shortt, John, ed. 1996, 1997. Charis Mathematics Units 1–9 and Units 10–19. St. Albans, United Kingdom: Association of Christian Teachers.
Toward a Christian Approach to Teaching and Learning Mathematics 5.4