Math Circle 1

Whither Mathematics Education in the 21st Century?

In the late 1960s and early 1970s, a math professor named Howard Eves wrote a series of books called In Mathematical Circles. He used the idea of a circle having 360 degrees to write 360 short pieces, each one showing something interesting about math—its beauty, history, people, humor, and real-life uses.

In a similar way, James Nickel is using that same approach to explore math from a Biblical Christian perspective, aiming to show its depth, meaning, and significance through the lens of faith.

It is my hope that the insights from our correspondence will find readers who are as fascinated by the wonder and beauty of mathematics as we are. If you’d like to become better acquainted with James Nickel, check out this introduction here.

Whither Mathematics Education in the 21st Century?

EN: In the essay, you contrast Morris Kline’s view of mathematics as humanity’s sole source of order in a meaningless universe with Cornelius Van Til’s and Johannes Kepler’s God-centered perspectives. How has this contrast shaped your conviction that mathematics education must reject the neutrality of knowledge, and what practical implications does this have for teachers today who might still rely on “neutral” textbooks?

JDN: Both Kline and Van Til/Kepler bring a framework into their understanding and doing life, life that also involves mathematics. No one can engage the discipline of mathematics presuming “neutrality,” for such unbiasedness is also a framework.

“Every point of view is a view from a point,” said one of my mentors, Dr. Glenn R. Martin (1935-2004). If the point of the way we view things is the revelation of who the Father is through Jesus Christ in the communion of the Holy Spirit, then all of life resounds with an entirely different harmony, a harmony that Dr. Kline could not hear because he began with meaninglessness.

So, we understand mathematics by first asking the question, “Who is God?” You may ask yourself, “Why would one start with a question like this? I thought math was just about doing sums and things like that. Why would ‘God talk’ have anything to do with counting and triangles?”

Georgetown University Professor James V. Schall (1928-2019) believes that this God question is “the most exciting and basic of all topics, the one that really gets to the heart of things, of why things are and why things are the way they are.” [The Order of Things (2007), p. 35.]

Schall, in the same book, later said [p. 59.], “… the paradigm of the order that we encounter in the world is already found in the Trinity of persons and their inner relation to one another.”

Since mathematics is all about structure (form) and order (the proper sequence to engage when solving problems involving mathematical equations and how the order of our minds interpenetrates the order of the world outside of our minds), it behooves us to understand the ground of such order.

A Trinitarian ground framework thus requires us to reinterpret everything “out there” written about mathematics, especially from math textbooks. Reinterpretation has been a driving factor in my life since the spring of 1979, when the aforementioned Dr. Martin challenged me to do so with prophetic imperative. There is a proliferation of math textbooks that are blind to the nature of reality and the Creator thereof. We do not make these books “Christian” by attaching Bible verses here and there. Reinterpretation is hard work, a diligent digging up the fallow ground of our minds with fresh inspiration of the Holy Spirit. Therefore, if a math teacher is relying on a textbook written by an author(s) void of framework thinking, let alone Trinitarian framework thinking, that fallow ground will be unturned, and the status quo of mathematics pedagogy will remain unchanged.

The many books and essays that I have written are my attempt to give a math teacher the tools that I have discovered in the past decades, implements by which to dig. It will take time to reorient one’s thinking and teaching, but the fruit is worth the effort.

EN: You propose three key reforms for teaching math: teaching skills and theory in creational context, strengthening the abstract/concrete connection, and incorporating science and history. Could you elaborate on one of these—perhaps the abstract/concrete link—and share an example from your own teaching or curriculum development (such as in The Dance of Number) of how tying abstract math to God’s creation transforms a student’s experience from “mathematics-lite” to “mathematics-heavy”?

JDN:There are two levels of abstraction involved in understanding and doing mathematics. The first abstraction, the Level 1 abstraction, is the transition from handling concrete objects like apples and attaching number symbols and relationships to their arrangement. Students can do this very early in life by encouraging them to group objects to make counting them easier, e.g., grouping apples by fives. (Note: they tend to do that without much encouragement!) If you can create three groups of five apples each and then have one more to count, you have 16 apples (5 + 5 + 5 + 1). The transition of sixteen apples into a mathematical statement involving symbols 5, +, 1, =, and 16, engages in a host of abstract concepts, i.e., the meaning of 5 (addend), 1 (addend), + (addition), = (equals), 16 (sum and place value!). The interplay of concrete and high-level abstract concepts is involved in writing this statement:

5 + 5 + 5 + 1 = 16

The second level of abstraction, Level 2, involves generalizing number relationships. Suppose you have a group of three apples and a group of five oranges. To determine the total amount of fruit, it does not matter if you take the count of apples first and then add the oranges to that count next, or vice versa. You get eight pieces of fruit either way. Level 2 abstraction is taking this situation and generalizing it with letters, letters that represent any number of apples and any number of oranges and then their sum when taken together, i.e.,

We let x = number of apples, y = number of oranges, and a = the sum of the two groups.

If you want to find the total number of pieces of fruit, i.e., a, it does not matter if you add apples to oranges or oranges to apples. We engage Level 2 abstraction involving letters for numbers (aka Algebra!). We write:

x + y = a and y + x = a or,

more easily, the equivalent x + y = y + x.

The student has now discovered a general rule involving addition, the rule that states that the order (You can start with x and then add y or start with y and then add x.) does not matter. The sum will be the same either way, provided you calculate the sums correctly!

An excellent resource in helping one see this concrete to abstract thinking in basic arithmetic is Ron Aharoni’s Arithmetic for Parents (2015).

Thinking from concrete to abstract in arithmetic is the necessary stage to prepare the mind for the type of next-level generality engaged by the world of Algebra, the language that mathematics speaks. The Scottish mathematical physicist James Clerk Maxwell (1831-1879) reflected on the nature of arithmetic, “Thus number may be said to rule the whole world of quantity, and the four rules of arithmetic may be regarded as the complete equipment of the mathematician.” [“Remarks on the Mathematical Classification of Physical Quantities,” in Proceedings of the London Mathematical Society, March 9, 1871.]

For Maxwell, arithmetic led him to his “mathematics-heavy” four equations that govern the interpenetration of electricity with magnetism, equations indispensable to our technological age of radio, television, the internet, smartphones, et al.

All of Maxwell’s work began with the framework that I explained in the answer to Question 1. He said, “… each individual man should do all he can to impress his own mind with the extent, the order, and the unity of the universe, and should carry these ideas with him as he reads such passages as the 1st Chap. of the Ep. to Colossians.” [letter to Bishop Ellicott, 22 November 1876. In Lewis Campbell and William Garnett, The Life of James Clerk Maxwell (1882), p. 191.]

EN: In the 21st century, are you seeing progress in the development of distinctly biblical curricula? What are the biggest challenges in this area?

JDN: Progress? Yes and No. For decades I looked for a curriculum that builds on a Trinitarian framework. I did not find any, so I wrote one: the four-volume work entitled The Dance of Number (2018) and then its John the Baptist introduction Mathematics: Exploring Beauty (2024). These books are a start that brings the challenge to others to work through and build upon, to carry the Trinitarian framework across the board, both in the multivariate world of mathematics and everything else that is lived and taught. I hope that the world of Classical Christian education, a pedagogy that engages the goal of goodness, truth, and beauty, will take up the torch of Trinitarian dynamics as the way to reinterpret these ancient Greek ideas, to impress them with the Maxwellian mindset, i.e., that the extent, the order, and the unity of the universe is grounded in the incarnate Son of the Father, who in scope both cosmic and personal, holds all things together (Colossians 1:15-17).

FOR MORE, VISIT https://biblicalchristianworldview.net/documents/whitherMathematics.pdf