Making it count: the importance of emphasizing mathematical concepts and positive math mindsets in elementary school

By Ariana Forsythe

Every senior at my school has to deliver and defend a senior thesis at the end of the year. I have previously shared the resulting work of one of my students. This year one of my students did a marvelous job in exploring mathematical mindsets. I am sharing (with permission) Ariana’s work below because 1) readers of this site may find it insightful, and 2) I’m just so darn proud of her.

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“Dear Maths. Here are 10 things I hate about you…With maths, you’re either right or you’re wrong. You can’t argue your way out of an incorrect answer….Maths teachers seemed angrier than all others…There seems to be no middle ground with maths. You’re either a maths person or you’re not. It’s unwelcoming. Exclusive. It doesn’t reach out” (Hunter). These are just a few of the reasons why blogger Kate Hunter, a self-proclaimed math hater who has allegedly never been good at math, has so much disdain for the subject. Her voice reflects those of countless Americans, as the phrase “I’m just not a math person” becomes a more and more prevalent excuse for poor math performance. In one tragic example, Joseph Cabral, now a successful freelance writer, was a straight-A student until he had to take algebra in middle school. Since then, he has taken and failed the course seven times throughout high school and college and has explained, “I started to question my character, my brain, my capabilities, and even my values” (Cabral). Countless American students like Joseph Cabral are struggling so much with math that they feel like failures, have limited career options, and hate the subject of mathematics.


Americans’ reputation for poor math abilities is not a recent development. Over a century ago, education experts were already noting Americans’ “meager results” in mathematics, and they have been disputing how to correct these results since then (Phillips). Throughout the 20th century, most traditional public education was dominated by the progressive ideals of John Dewey and William Heard Kilpatrick who believed worthwhile information was that which was useful in everyday life or for a student’s specific career interests. This more pragmatic but less academic strategy backfired during World War II when army recruits had such poor understanding of basic arithmetic for bookkeeping and gunnery that math education had to be added to military training (Klein). By the Cold War, math education reform was popularized, and a model called “new math” was pushed throughout American classrooms in attempts to improve students’ abilities to grasp concepts through abstract thinking rather than by memorization of facts (Phillips). In 1958, the American Mathematical Society founded the School Mathematics Study Group which led to math as advanced as calculus being incorporated into high school curriculums (Klein). In the seventies, however, education professionals noticed declining math test scores as a result of the new math model, so curriculum returned to the more traditional, simplified, memorization-centered approach (Philips). By the nineties, a so-called “math war” arose between the U.S. Department of Education and university mathematicians. Since then, professionals have continued their efforts to reform standardized tests, skills requirements, curriculum, and teacher training in efforts to raise Americans’ scores (Klein).

Nevertheless, American adults and children continue to demonstrate meagre math abilities on the international level to this day. Adults between the ages of 16 and 65 from 23 different economically advanced countries were surveyed in 2012 by the Organization for Economic Cooperation and Development in a test called the Program for the International Assessment of Adult Competencies, or the PIAAC. On the math portion of this survey, American adults scored 16th out of the 23 participating countries, with the average American score 26 points below the average of the highest scoring country, Japan, and only 20 points above the average of the lowest scoring country, Italy (Literacy, Numeracy, and Problem Solving in Technology). Similarly, when tested amongst 15-year-old international high school students, Americans scored 35th out of 70 participating countries and 13 points below the international math average on the Programme for International Student Assessment, or PISA (PISA 2012 Results in Focus). While the United States is a leader in world economics and politics, it continues to lag behind educationally and perform below average in mathematics, as seen in both children and adults.

A common psychological phenomenon relevant to the discussion of education reform concerns the debate about whether intelligence is genetically fixed or acquired through effort. Is it nature or nurture? For a long time, the prevailing conception was that genetics determined brain power, as seen in geniuses like Einstein and Beethoven, but scientists such as Bruce Wexler, a neuroscientist and professor of psychiatry at Yale University School of Medicine in New Haven, recently confirmed that any genetic predispositions from birth are surpassed by superior education and effort (Boaler 5). While many Americans see their high schoolers underperforming in mathematics on a global scale and conclude that they are genetically predisposed to struggle with math, studies have revealed that proper encouragement can greatly increase students capabilities (Kimball). This is evident in PhD Carol Dweck’s idea of “the mindset” as developed in her 2006 book Mindset: The New Psychology of Success. Through her research, Dweck discovered that people’s mindsets, or perceptions of themselves and the world around them, dictate their success. Those with fixed mindsets believe that they have fixed potential; to them, challenge means inadequacy, and failure reflects directly on their person. Those with growth mindsets know they have untapped potential; they know they can always keep growing, so they know that challenges will eventually lead to success (Dweck). For over a century, America has lagged behind in mathematics achievement and continues to demonstrate poor math abilities on international tests, despite being a great and prosperous international power. There is now substantial scientific proof, however, that genetics have little to do with students’ intellectual abilities, but learning experiences do. Past educational reforms have dealt primarily with curriculum reform, but one important component in math education that has not been fully addressed is the mindset students have about math.

The argument for math

Americans’ poor performance in mathematics needs to be addressed in elementary education before it can be improved at the secondary level. This math crisis should be addressed earlier because the foundations of basic conceptualization necessary for higher-level, abstract, mathematical reasoning are set at this time. Furthermore, children between the ages of 4 and 10 have the highest brain activity of their lives and are mentally equipped to absorb not only procedural but also conceptual teachings of elementary school. Unfortunately, students’ mindsets about math are also set at this time, and they are often discouraged by teachers not specialized in mathematical instruction. Ultimately, elementary schools need to employ math specialists to teach elementary math classes or, at a minimum, to instruct generalist teachers on the latest and most effective strategies of presenting math curriculum in constructive and positive ways.

Students are currently lacking adequate foundations in basic math concepts that they need to understand algebra and higher quantitative reasoning. As mentioned earlier, on the most common international test taken by 15-year-old high school students, the PISA, the U.S. scored 35th out of the 70 participating countries and 13 points below the average (PISA 2012 Results in Focus). On a different type of test, the Trends in International Mathematics and Science Study, or TIMSS, American fourth graders ranked 14th out of the 49 participating countries and 39 points above the average on the 2015 math section (“Mathematics for Grades 4 and 8: Averages”). American students are performing competitively on international tests at younger ages but are scoring below average by the time they reach high school, a trend directly opposite to higher scoring countries. One possible explanation is that American schools are teaching students for immediate success through math facts, memorization, and procedural approaches but not amply preparing them for the abstract reasoning required in subsequent study of higher level math that begins in middle school, because elementary math education does not emphasize mathematical concepts as a stepping stone to abstraction. In Singapore, for example, elementary students are required to use problem solving and visual models to learn mathematical concepts while American students are primarily taught procedures for obtaining similar results. The problem solving approach is initially more confusing and challenging, but the concepts learned through problem-solving are the building blocks for abstraction, which ultimately prepares students better for higher level mathematics than a procedural approach.

This defense for strong conceptual foundations is discussed in the book Mind, Brain, and Education Science by Tracey Tokuhama-Espinosa, PhD and Director of the Institute for Teaching and Learning at San Francisco University of Quito. Her research shows how children need points of reference to connect to new information and how this means that poor instruction early on will continue to cause students to fail in math, a cumulative subject. She says, “Without a firm foundation in basic mathematical conceptualization… a student will have a lot of trouble moving on to build more complex conceptual understandings” (Tokuhama-Espinosa 35). All students must make this jump from concrete to abstract—from arithmetic to algebra and beyond—but American students have not been given the tools in elementary school to make this leap smoothly.

Another reason to focus stronger math instruction toward elementary students is their comparatively greater brain activity at this age. In the 1980s, Harold Chugani was the director of the PET Center at the Children’s Hospital of Michigan and used developing technology to record children’s brain activity at different ages. He found that 4-year-old brains consume copious amounts of glucose and are twice as active as adult brains. This extremely high level of activity continues until age 10 and gradually decreases until about age 16, when brain activity levels to that of adults. Children really are sponges for information at this early age, and as Michael Phelps, a biophysicist at UCLA, not the Olympic swimmer, who develops and researches with neuroscience technologies, explains, “If we teach our children early enough, it will affect the organization, or ‘wiring,’ of their brains” (Nadia). This easily applies to the instruction and development of math concepts for elementary students and demonstrates how math education is critical in the early years for setting up success later.

Similarly to how elementary math education sets the conceptual foundation for higher level mathematics, elementary math education also shapes the mindsets students have about the subject. Unfortunately, the current education system fosters negative mindsets about mathematics, especially when students at young ages are indirectly taught that mistakes equal failure. Dr. Jo Boaler, author of Mathematical Mindsets, recounts a lecture with Stanford professor Li Ka Shing who explained, “Every time a student makes a mistake in math, they grow a synapse” (Boaler 11). These synapses occur both when a student first makes a mistake—when the answer is not reconcilable with prior experiences—and when an outside source, or teacher, corrects the mistake. Accumulations of synapses lead to brain growth, meaning that mistakes in math also lead to brain growth. Because of this, teachers should treat their students’ mistakes as opportunities for instruction and growth (Boaler 11). In reality, mathematics is a diverse, creative subject as there are countless, different, equally correct ways to approach each problem, yet math continues to be taught from the perspective that there is only one correct answer. Unfortunately, field researchers like Dr. Jo Boaler believe American students are too often discouraged from making mistakes, from freely exploring different solutions, and from joyfully embracing math (Boaler).

One solution to this issue of negative mindsets is to employ math specialists, who are more knowledgable on how best to teach math and foster their students’ love of the subject. Francis Fennell, former president of the National Council of Teachers of Mathematics, explained that generalists usually take “two or three courses in mathematics content and one course in the teaching of mathematics” and that “their teaching load generally consists of a full range of subjects, with particular attention to reading or language arts in a self-contained classroom” (Fennell). On the other hand, math specialists are required to take more math courses and have more math teaching experience in addition to the requirements of generalist teachers. In Arizona, for example, endorsed math specialists must have an “elementary or special education teaching certificate, 3 years in full time K-8 teaching experience…18 semester hours of courses in mathematics content, 3…in mathematics classroom assessment, and 3…in research-based practices, pedagogy, and instructional leadership in mathematics” (“Mathematics Specialist Certifications (by State) with Descriptions”).What is needed, then, are math specialists who can keep up with the constantly evolving and perfected mathematical teaching approaches, who know ways to teach to different kinds of learners, and whose love of math can inspire their students. Math specialists can be actively involved in the on-going discussions of how best to teach math with time that generalists simply do not have, as they must be well rounded in several areas of study. If Americans want to remedy poor performance in math, they need to call for math specialists to properly lay foundational, conceptual knowledge and foster positive mathematical mindsets at an early age for their students.

The case against math

Some skeptics argue that America, on the whole, is not actually performing poorly but that the scores of low scoring students are dragging down those of high scoring students, affecting America’s average math test score. The results from the 2012 PISA international math test revealed that 15% of American’s score variances, or the difference between high and low scoring students, was due to socio-economic status, a percentage twice as large as those of high scoring countries such as Finland, Japan, and Hong Kong (Ryan). America is an extremely diverse nation, especially compared to countries such as Finland or Japan, so some experts believe that whatever attempts are made to close this achievement gap or raise overall scores will always be futile or will always leave some students behind.

While it is true that part of Americans’ poor results on the PISA math test is due to diversity that many other participating countries do not have, it is important to address another perspective that the data shows: resiliency. This is when economically disadvantaged students from the bottom 25% income bracket perform better than predicted, or overcome their negative odds. The U.S. has a lower than average percentage of resilient students (Ryan), which can be attributed to the overall negative mindsets about mathematics that leads to complacency. Ethnic minority and socio-economically disadvantaged students especially often have more difficulties excelling academically. This is largely due to a phenomenon experts call “stereotype threat,” or when teachers’ and administrators’ intentional or unintentional racist or sexist stereotypes negatively impact students. America as a whole needs a mathematical mindset shift, and math specialists are studying how to address social stigmas in reforming cultural mindsets to help close the achievement gap.

Some people, especially parents of struggling students, argue that many students are incapable of comprehending higher level math and that increased efforts are unnecessary since people can still find career success without advanced math understanding. One very prominent, passionate voice, Andrew Hacker, describes how millions of high school and college students are struggling with math courses such as algebra and asks, “Why do we subject American students to this ordeal?” (Hacker). He argues that supplementary courses such as finance and economics should be offered for students who do not have the natural aptitudes needed to complete algebra. Hacker is not alone in his belief that math is holding students with poor math aptitudes back from their academic goals. He cites Barbara Bonham of Appalachian State University who has seen students take and retake algebra up to five times without passing (Hacker). Their proposal to solve the American math crisis is to lower math course requirements so that students with poor math abilities can graduate from high school and college.

However, the issue lies not with genetic abilities but instead with how the different students were prepared differently for a challenging but doable course. As PhD Tracey Tokuhama-Espinosa explains in her book, students who are struggling in algebra and high school level math courses are encountering issues due to the crumbling foundation of elementary math education because math is a cumulative subject (Tokuhama-Espinosa), for if their misconceptions are not properly address early on, they will never be corrected and will continue to fail. As seen during World War II, when soldiers did not even understand enough mathematics for bookkeeping and gunnery, a pragmatic approach to math education only perpetuates Americans’ poor math skills on the international stage (Klein). Though many students may not need algebra or calculus for their music or social-work careers, that is not a reason to teach math poorly and ineffectively. Furthermore, giving students strong mathematical foundations in elementary school can decrease the number of students who are deficient in math.


Americans need to demand reforms to math education, specifically in elementary school. Children’s brain activity during this time allows them to be informational sponges. They are well-equipped to absorb complex conceptual understandings, but this also means they are easily susceptible to contracting negative mindsets about mathematics. When elementary math teachers can foster positive mathematical mindsets in their students, we will see classrooms filled with children actively working together and problem solving in pursuit of a common goal. Dear Math. Here is why I love you. With math, there are countless correct ways to get to the right answer. You can collaborate to solve problems. And math teachers, they help me to love math. Anyone can be a “math person.”

Works Cited

Boaler, Jo. Mathematical Mindsets: Unleashing Students’ Potential through Creative Math, Inspiring Messages, and Innovative Teaching. San Francisco, CA, Jossey-Bass & Pfeiffer Imprints, 2016.

Cabral, Javier. “How Algebra Ruined My Chances of Getting a College Education.” KCRW, 2 May 2013,         college-education. Accessed 26 Jan. 2017.

Dweck, Carol S. Mindset: The New Psychology of Success. Ballantine Books, 2008.

Fennell, Francis. “We Need Elementary School Mathematics Specialists NOW.” National Council of Teachers of Mathematics, 2006, Accessed 3 Apr. 2017.

Hacker, Andrew. “Is Algebra Necessary?” The New York Times, 28 July 2012,                Accessed 10         Dec. 2016.

Hunter, Kate. “Dear Maths. Here Are 10 Things I Hate about You.” MamaM!a, 5 Sept. 2012, Accessed 26 Apr. 2017.

Kimball, Miles, and Noah Smith. “The Myth of ‘I’m Bad at Math’.” The Atlantic, Atlantic Media Company, 28 Oct. 2013, Accessed 28 Oct. 2016.

Klein, David. “A Brief History of American K-12 Mathematics Education in the 20th Century.” California State University, Northridge. Ed. James Royer. Information Age Publishing,   2003. Accessed 18 Oct. 2016.

“Mathematics for Grades 4 and 8: Averages.” Trends in International Mathematics and Science Study (TIMSS), National Center for Education Statistics, 2015, Accessed 3 Apr. 2017.

“Mathematics Specialist Certifications (by State) with Descriptions.” Elementary Mathematics Specialists & Teacher Leaders Project. N.p., 2009-2015. Web. 19 Apr. 2017.

Literacy, Numeracy, and Problem Solving in Technology- Rich Environments Among U.S. Adults: Results from the Program for the International Assessment of Adult Competencies 2012. National Center for Education Statistics, Oct. 2013. Accessed 20 Apr. 2017.

Nadia, Steve. “Riggs News.” The Riggs Institute, Accessed 3 Apr. 2017.

Phillips, Christopher J. “The Politics of Math Education.” The New York Times, 2 Dec. 2015, Accessed 15 Oct. 2016.

PISA 2012 Results in Focus. Organisation for Economic Co-Operation and Development, 2014, Accessed 3 Apr. 2017.

Ryan, Julia. “American Schools vs. the World: Expensive, Unequal, Bad at Math.” The Atlantic.   Atlantic Media Company, 3 Dec. 2013. Accessed 20 Oct. 2016.

Tokuhama-Espinosa, Tracey, and Judy Willis. Mind, Brain, and Education Science: A             Comprehensive Guide to the New Brain-Based Teaching. W.W. Norton, 2011.

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

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).


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


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.


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