Love should be the goal of education

by Jacob Mohler

The following post comes from Jacob Mohler, Math department co-chair at Westminster Christian Academy in St. Louis, Missouri. From Jacob’s bio on the Westminster website:

I became a math teacher because I wanted students to learn math in a better environment than I did. A teacher who was trained to teach English taught me math. There are wonderful notions in math of how numbers found in the natural world point to a Creator. For this reason, as I noticed how God has left clues for us to see His handiwork, I wanted to find ways to show students similar things. Thinking about math as the language and logic God used to create the world was too good a secret for me to keep. Now I want students to think about their involvement in the mathematical enterprise as a means to join God as co-creators of interesting things.

Mathematics allows humans to make the “invisible” part of the created world become “visible” in a way to tame it, describe it, use it, wonder about it, write about it, have control over aspects of it. These ways of making the “invisible world become more visible” suggest a knowable and dependable world that is worth knowing and caring for to enhance the human experience. Furthermore, from the perspective of a Christian in the stream of the Reformation tradition, one might say that humans’ ability to know the world in these ways is “thinking God’s thoughts after Him” and living out our Imago Dei as creators, nay, co creators, as we do not Create from nothing as God did.

Humans endeavoring to learn math from our forerunners is necessary to continue the timeless knowledge of our past, but we must not think that simple memory of facts and procedures will guarantee that we will learn what is needed to be faithful messengers to the next generation or that we can rightly use the past understanding of mathematics to “better the affairs of mankind” or as Sir Francis Bacon so aptly put it, “relief of man’s estate.” Learning mathematics should be seen as a work in progress as a training and maturing and not as mechanical or a behavioral modification that tests learning by simple reciting of known and recognized nomenclature, although memorizing of agreed upon facts might be the minimum we ask of students, we need to stress that knowing the world of math is not the same as duplicating what a teacher shows.

The important facet of thinking about mathematics as a training and maturing is that each student’s background of mathematics, abilities, ways of thinking and even difficulties with learning ALL influence how teachers decide to design class lessons and assessments. Routine and simple problems are set in front of students as well as non routine and challenging ones. Whether in a public, private, parochial or protestant christian school the teaching and learning of mathematics may look similar because knowledge of math is similar to knowing God’s world in general terms. This is often referred to as general revelation. All humans have access to this form of knowing about the world.

The depth of knowing about the world is not the same for each person, partly because of interest in knowing, time committed to learning and personal abilities differ. Here is where notions of giftedness bumps up against various ways we as humans interact with the various aspects of enjoying the world. Some people are more athletic than others, for example, and some will be better surfers than others. This reality should not affect the the idea that humans were meant in the Imago Dei and can interact and know the world in small, large, and enjoyable ways. Just because I will never be an ukulele player invited to in Carnegie Hall does not mean that I should not work to improve my skills to bring enjoyment to me and others in my spheres of influence. In a similar way, learning subjects beyond personal giftedness or interest should be encouraged because learning broadly betters each person’s ability to interact in the natural world by enjoying it and the more people know the more they have ability to love more broadly.

Love should be the goal for education.

The Role of Mathematical Aesthetics in Christian Education

by R. Scott Eberle

Scott Eberle has a Ph.D. in Math Education and currently serves as a missionary in Niger, working to spread the Gospel message through Christian education. Scott works to build up Christian leaders and educators in Niger who approach mathematics through a distinctly Christian perspective. You can follow Scott and his family at nigerministry.tumblr.com.

Josh Wilkerson invited me to contribute something on the aesthetics of mathematics from a Christian perspective. I’d especially like to discuss how such a seemingly abstract idea has application in Christian education.

Detail of the Mandelbrot Set in the plane of complex numbers
Detail of the Mandelbrot Set in the plane of complex numbers

Mathematical Aesthetics

Mathematics has been considered an aesthetic subject from antiquity. The Greeks considered mathematics to be the highest form of aesthetics because of its perfection. The Pythagoreans and Platonists considered mathematical concepts to have a real, mystical existence in some perfect realm.

Throughout history, mathematicians and philosophers have continued to claim that mathematics is beautiful for a variety of reasons. For example, whereas the Greeks saw beauty in the ontology of mathematics, the French mathematician Henri Poincaré saw beauty in its epistemology. Because of the way we teach mathematics, many students believe there is always one hard-and-fast method for cranking out the answer to any mathematical problem. But as mathematicians know, true mathematical problems require a great deal of creative intuition to solve. Poincaré pointed out that mathematicians rely on aesthetic-based intuition to distinguish fruitful paths of mathematical inquiry from dead ends. He wrote, “It is this special aesthetic sensibility which plays the rôle of the delicate sieve” (1908/2000, p. 92).

Today, nearly all mathematicians continue to recognize the aesthetic nature of mathematics (Burton, 1999). The British mathematician John Horton Conway went so far as to claim, “It’s a thing that non-mathematicians don’t realize. Mathematics is actually an aesthetic subject almost entirely” (Spencer, 2001, p. 165). The reason the general population doesn’t realize that mathematics is an aesthetic subject is probably due to the mechanical way in which we frequently teach mathematics. School exercises are often artificial, simplistic, and have only one right answer. There is nothing creative or aesthetic to see in the average math lesson.

Conway’s claim that mathematics is almost entirely aesthetic is a bold one. But actually, modern mathematics can be seen to be an aesthetic subject from its foundations to its methods to its end results. This is especially true since mathematics’ divorce from physics in the 19th century as it became a purely abstract study, inspired by, but independent of, the natural world.

  • Foundations: Mathematics rests on a foundation of axioms and definitions. But these are chosen, not deduced. Mathematicians choose definitions and axiomatic systems based on criteria of logic, relative completeness, consistency, mutual independence, simplicity, connectedness, and elegance. These criteria are partly aesthetic in nature.
  • Methods: As Poincaré pointed out, mathematicians rely on a certain aesthetic sense to guide their explorations. Paths that seem particularly elegant often prove to be the most successful. In 1931 Gödel destroyed earlier hopes of purely mechanical methods of generating mathematical theorems and proofs, making the fundamental role of intuition even more necessary. Modern researchers are beginning to understand that intuition is not a fuzzy feeling, but rather a rigorous source of insight. Robert Root-Bernstein (2002) makes a powerful argument that all scientific thought occurs first as an aesthetic intuition, and is then confirmed by verbal logic. Therefore aesthetics guides our mathematical exploration and is the basis for our mathematical reasoning. But we often show only the final algorithmic logic to our students.
  • Results: Mathematicians don’t often discuss aesthetics explicitly, but when they do, they usually point to theorems and proofs, which they insist should be elegant. The American mathematician Morris Kline observed that “Much research for new proofs of theorems already correctly established is undertaken simply because the existing proofs have no aesthetic appeal” (1964). Mathematicians especially appreciate results which are surprisingly simple or have significant connections or visual appeal. Such results are said to be beautiful. The Mandelbrot Set, for example, is beautiful partly because its definition is surprisingly simple and partly because it has great visual appeal. It is interesting to note that criteria such as significant connections indicate that beautiful results will be among the most useful and important. Criteria such as surprise suggest that beautiful results may be important for insight and understanding, and therefore also for education.

So mathematics is seen to be aesthetic “almost entirely.” At this point, some would say the discussion is merely philosophical and has no real world implications. Indeed, most mathematicians give aesthetics little explicit thought unless questioned about it. It is perhaps for this reason that many educators have not picked up on the importance of aesthetics in mathematics.

Simple, visual “proof” of the Pythagorean Theorem by Bhaskara II (12th century AD)
Simple, visual “proof” of the Pythagorean Theorem by Bhaskara II (12th century AD)

The Christian View

Throughout history, most mathematicians have been Platonist, at least in practice. We tend to think that mathematical ideas are discovered, rather than invented. In more recent times, some have questioned this, claiming that mathematics is simply the brain’s way of understanding how the universe is structured, and mathematics could be very different for an extraterrestrial species. (See, for example, Lakoff & Núñez, 2000.) Others disagree, pointing out how mathematics inexplicably predicts new discoveries. Of course, all agree that certain things, such as notation, conventions, and choice of axioms, are man’s invention. But where do the beautiful results we admire come from? The Greeks cannot be said to have “invented” the Pythagorean Theorem. Most would agree they (and other cultures) “discovered” it.

Most Christian theologians, from Augustine (354 – 430 AD) onwards, as well as Christian mathematicians, have agreed with a Platonist perspective, believing that mathematics is in the mind of God, and we discover these eternal truths. Mathematics cannot be part of Creation because it is not a physical part of nature—it is a collection of abstract ideas. One does not physically create abstract ideas, one conceives them. And God must have always known these ideas, so they have always been part of his thoughts. Mathematics preceded Creation and is untouched by the Fall. It is perfect and beautiful and contains awe-inspiring ideas, such as Cantorian infinity, which is part of God’s nature but not part of our physical universe. However, we ourselves are fallen, so our understanding and use of mathematics is imperfect.

Some modern Christian thinkers have proposed other possibilities similar to those of Lakoff and Núñez, making mathematics a human activity, or only one of many possible systems of mathematics in the mind of God. Nevertheless, all Christians affirm that mathematics is not independent of God. Even if there are other possible systems of mathematics, the one we know is the one God chose for us as good, and it has always been known by God. It is not some arbitrary invention. I like to think that when I am studying mathematics, I am studying the very thoughts of God, that mathematics is part of God’s attributes. God did not “create” love; God is love (1 John 4:8). Likewise God did not “create” one and three; God is one Being in three Persons. God did not “create” infinity; God is infinite. And so on.

But whatever position you take, whatever the ontology of mathematics, it should not surprise us that mathematics is beautiful, because God is beautiful. Mathematics is indeed “an aesthetic subject almost entirely.” Mathematical beauty and usefulness is a mystery only if we do not believe it comes from God. (See, for example, the classic article by Wigner, 1960.)

Euler's Identity, relating five fundamental constants and three basic operations, is often called the most beautiful result in mathematics (Wells, 1990).
Euler’s Identity, relating five fundamental constants and three basic operations,
is often called the most beautiful result in mathematics (Wells, 1990).

Education

Though aesthetics is part of the very foundation of mathematics, it is largely neglected in math classrooms. As mathematician Seymour Papert pointed out, “If mathematical aesthetics gets any attention in the schools, it is as an epiphenomenon, an icing on the mathematical cake, rather than as the driving force which makes mathematical thinking function” (1980, p. 192). However, an increasing number of researchers (including myself) have been noting important consequences of mathematical aesthetics for how we teach mathematics at all ages.

The interested reader can turn to researchers such as Nathalie Sinclair to see how modern research has been discovering the importance of aesthetics in mathematics education. Aesthetics is a “way of knowing” mathematics prior to verbal reasoning and should be an important part of our mathematics classrooms. Indeed, Sinclair (2008) has found that good math teachers tend to use aesthetic cues in their teaching implicitly, though they may not realize it. For example, teachers who reveal a “secret weapon” or present a surprising fact or note simpler ways to express certain solutions are modeling a useful aesthetic to their students. In my own research (Eberle, 2014), I have found that even elementary school children come with their own aesthetic ideas and use them in valid mathematical ways when given the opportunity to do open-ended math problems. And this is true of all children, not just those that are gifted in mathematics. Children’s initial aesthetic ideas are far from those of mathematicians, but through experience they are refined. Educators from John Dewey to the present day have argued that aesthetics is important for all of education, and now we are discovering how this is true for mathematics.

Nathalie Sinclair (2006) has proposed that mathematical aesthetics has three roles in education:

  1. Aesthetics gives intrinsic motivation to do mathematics. This is in contrast to the extrinsic coaxing we often use with students. Instead of “sugar-coating” math problems by placing them in artificial contexts, we should allow students to explore the natural symmetry and patterns found in every branch of mathematics. I sometimes challenge teachers to see how many patterns they can find in the “boring” multiplication table. They are usually very surprised. Students can also engage in mathematics in a natural way by pursuing projects they themselves have suggested. Such genuine contexts are highly motivational. (See these posts by Josh Wilkerson for a Christian perspective on this idea.)
  2. Just as with mathematicians, aesthetics guides students to generative paths of inquiry. When allowed to explore freely, children use their own aesthetics to find valid mathematical insights, though this may take time. Students need opportunities to pursue their own ideas and conjectures.
  3. Aesthetics helps students to evaluate their results. Often math is presented as black-and-white with only right and wrong answers. But if students are allowed to do more open-ended inquiry or project-based mathematics, they can use their growing sense of aesthetics to evaluate the solutions found.
Solution to an open-ended geometry problem found by a 4th grader by using aesthetic symmetry
Solution to an open-ended geometry problem found by a 4th grader by using aesthetic symmetry

Christian Education

As Christian educators, we should realize that God gives common grace and we should always be open to learning from the best results of secular research, filtered through the worldview shaped by our faith. Throughout history, Christians have often been at the forefront of recognizing the importance of aesthetics. God gave us our ability to appreciate beauty and patterns for a reason, and what is math if not the study of patterns (Hardy, 1940)? We Christians should be among the first to recognize the importance of educating the whole child, even in mathematics, and embracing research showing the importance of allowing aesthetics to have a deep role in education, including our mathematics instruction.

Even more importantly, we should be careful not to make a sharp dichotomy between “secular” knowledge and “spiritual” knowledge. Mathematics is often taught as if our faith had nothing to do with the knowledge we are learning. Though it is wrong to artificially “spiritualize” every lesson, at the very least Christian students should understand the relationship between their faith and their studies. One way to do this is to let students know that math is not just a series of arbitrary algorithms and heuristics to be memorized, but a rich, creative, beautiful subject to be explored and appreciated. And when students see some of the beauty of the subject, we can lead them to reflect on the Source of that beauty. Indeed, we are doing a great disservice to Christian students if we lead them to believe that a subject that is in the mind of God is somehow boring or ugly.

I have to admit I am distressed sometimes by certain popular views of mathematics. I remember reading one author who wrote that mathematics was part of Creation, and as such, the author seemed to believe mathematics was purely arbitrary, as if there were no special reason God created 2 + 2 to be 4. I often come across this idea that math is not understandable, a result of learning by rote. All we can supposedly do is grit our teeth and memorize the mysterious methods. This author’s solution was to teach students to plug away at exercises and learn to praise God every time they correctly found God’s answer, and be thankful that God, in his faithfulness, had not changed the answer in the meantime. I fear that such instruction will not generate praise for God but rather fear of mathematics. My hope is that we can learn instead how to teach that mathematics is a deep, joyful, meaningful, beautiful subject. It is a reflection of God’s nature.

Flower with spirals in Fibonacci sequence Helianthus flower, Bannerghatta Bangalore by L. Shyamal / CC-BY-2.5
Flower with spirals in Fibonacci sequence
Helianthus flower, Bannerghatta Bangalore by L. Shyamal / CC-BY-2.5

Conclusion

For Christians, mathematical aesthetics must not be an optional extra-credit topic, but must rather be at the very foundation of our mathematics teaching. As Christian educators, aesthetics should guide our understanding of mathematics, inform the way we teach, and be a goal for our students’ learning—and this from the youngest ages. Just as students learn to appreciate poetry or music, Christian students should learn that mathematics is beautiful, and why.

References

Burton, L. (1999). The practice of mathematicians: What do they tell us about coming to know mathematics? Educational Studies in Mathematics, 37(2), 121-143.

Eberle, R. S. (2014). The role of children’s mathematical aesthetics: The case of tessellations. The Journal of Mathematical Behavior, 35, 129-143.

Hardy, G. H. (1940). A mathematician’s apology (1967 with Foreword by C. P. Snow ed.). Cambridge, UK: Cambridge University Press.

Kline, M. (1964). Mathematics in Western Culture (Electronic version ed.). New York: Oxford University Press.

Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books.

Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. New York: Basic Books.

Poincaré, H. (2000). Mathematical creation. Resonance, 5(2), 85-94. (Original work published 1908)

Root-Bernstein, R. S. (2002). Aesthetic cognition. International Studies in the Philosophy of Science, 16(1), 61-77.

Sinclair, N. (2006). Mathematics and beauty: Aesthetic approaches to teaching children. New York: Teachers College Press.

Sinclair, N. (2008). Attending to the aesthetic in the mathematics classroom. For the Learning of Mathematics, 28(1), 29-35.

Spencer, J. (2001). Opinion. Notices of the AMS, 48(2), 165.

Wells, D. (1990). Are these the most beautiful? The Mathematical Intelligencer, 12(3), 37-41.

Wigner, E. (1960). The unreasonable effectiveness of mathematics in the natural sciences. Communications in Pure and Applied Mathematics, 13(1).

Math for Goodness Sake

The following is from a presentation I gave at the 2014 Society for Classical Learning Conference. The title of my talk was “Math for Goodness Sake.” Here is the abstract:

The mission of Regents School of Austin (and undoubtedly any classical Christian education) is to equip students to know, love and practice that which is true, good and beautiful. How does the teaching of mathematics fit into this mission? There are numerous resources that address the beauty and truth of mathematics, but how do we instill in students an appreciation for the inherent goodness of math (especially since many students experience mathematics as confusing, stressful and generally contrary to anything considered good)?

Mathematics is at its core a good and virtuous activity and its enjoyment is not reserved for the “intellectual elite.” This talk will examine how the goodness of mathematics is actually rooted in a proper understanding of the ordered nature of God. When viewed through this lens we can see mathematics as a missional activity that increases our versatility as worshipers of Christ. Several practical methods for developing what I term “mathematical affections” will be presented for consideration and classroom use.

You can find a full list of presenters and the topics discussed in the 2014 SCL Conference Program. To get a further idea of some of the great discussion the SCL has going on the teaching of mathematics specifically I highly recommend checking out the SCL Journal, Fall 2013 – Moving Beyond Mechanics: Teaching Math Classically.

Below you will find my PowerPoint presentation (simply click the image for the link) and the written outline of my talk. I believe SCL will be recording presentations, so hopefully I will eventually be able to post (or link to) the audio as well.

UPDATE (8/15/14): Here is a link to the audio. You may need to become a member of the Society for Classical Learning website in order to access the linked audio, though I am not sure. 

Math for Goodness Sake Outline.

math for goodness sake

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