What is math? It might seem obvious: We hope we know what we're teaching — and that our students know what they're learning! But responses to that question can be surprisingly diverse.

Before you read on, please take a few minutes to reflect and record your own thoughts on “What is math?”

Illustration courtesy of Wendy Petti. |

I'm a math educator — I teach grade 4 math, I've created a math Web site, and I write about topics in math education. Yet I cannot easily express what math is. I'm in good company.

Bertrand Russell has quipped, “Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.” Math teacher Sanderson M.

Smith observes that “students can gain a tremendous appreciation for mathematics if they understand that the question 'What is mathematics?' has been analyzed and debated since the time of the Pythagoreans, around 550 B.C. “

It is worth pursuing a clear understanding of the meaning and scope of mathematics so that we might provide our students with a richer learning experience and help them more fully appreciate the beauty and power of mathematics.

## STUDENTS AND TEACHERS REFLECT ON “WHAT IS MATH?”

## What is Mathematics?

Mathematics is an old, broad, and deep discipline (field of study). People working to improve math education need to understand “What is Mathematics?” |

Mathematics as a formal area of teaching and learning was developed about 5,000 years ago by the Sumerians. They did this at the same time as they developed reading and writing. However, the roots of mathematics go back much more than 5,000 years.

Throughout their history, humans have faced the need to measure and communicate about time, quantity, and distance. The Ishango Bone (see ahttp://www.math.buffalo.edu/mad/ Ancient-Africa/ishango.html and http://www.naturalsciences.be/expo/ishango/ en/ishango/riddle.html) is a bone tool handle approximately 20,000 years old.

The picture given below shows Sumerian clay tokens whose use began about 11,000 years ago (see http://www.sumerian.org/tokens.htm). Such clay tokens were a predecessor to reading, writing, and mathematics.

The development of reading, writing, and formal mathematics 5,000 years ago allowed the codification of math knowledge, formal instruction in mathematics, and began a steady accumulation of mathematical knowledge.

### Mathematics as a Discipline

A discipline (a organized, formal field of study) such as mathematics tends to be defined by the types of problems it addresses, the methods it uses to address these problems, and the results it has achieved. One way to organize this set of information is to divide it into the following three categories (of course, they overlap each other):

- Mathematics as a human endeavor. For example, consider the math of measurement of time such as years, seasons, months, weeks, days, and so on. Or, consider the measurement of distance, and the different systems of distance measurement that developed throughout the world. Or, think about math in art, dance, and music. There is a rich history of human development of mathematics and mathematical uses in our modern society.
- Mathematics as a discipline. You are familiar with lots of academic disciplines such as archeology, biology, chemistry, economics, history, psychology, sociology, and so on. Mathematics is a broad and deep discipline that is continuing to grow in breadth and depth. Nowadays, a Ph.D. research dissertation in mathematics is typically narrowly focused on definitions, theorems, and proofs related to a single problem in a narrow subfield in mathematics.

## What is Mathematics? in the US

The study of mathematics is a popular subject in the U.S. International students interested in studying math might want to know which subjects are covered in most programs in the U.S. Typical areas include algebra, calculus, and geometry among others.

Mathematics is a popular and vibrant area of study in the U.S. But it is difficult to say precisely what is math. That is in part because mathematics is used for so many different ends.

A rough general definition is that mathematics is the study of the relationships between numbers, quantities, magnitudes, and other aspects of abstract reality. If you are an international student interested in studying math in the U.S.

, you might be particularly interested in which subjects within mathematics are typically covered in university courses.

### Algebra

Along the way to earning a math degree, every student will be expected to have a thorough understanding of algebra. In studying algebra, international students will learn about numerical operations and functions.

Many times this will involve executing operations in the correct order to find a solution or solve an equation. Many students will be exposed to algebra in elementary and secondary education.

In university courses, more complex and difficult branches of algebra are covered, including linear and abstract algebra.

### Calculus

Virtually all programs require that students be well versed in calculus to earn a math degree, as well. Calculus is useful for studying processes under variable time changes (such as in studying acceleration), complex slopes, and more complex functions than are dealt with in algebra.

There are two broad types of calculus: differential and integral calculus. Differential calculus involves determining the rate in which quantities change over time with respect to independent variables. The newer function is called the derivative function and the process of determining what that function is known as differentiating. That is how differential calculus gets its name.

Integral calculus is the study of two sorts of integrals, indefinite and definite. The indefinite integral is the antiderivative and involves finding a function from a derived function, as described in the previous paragraph. The definite integral is called the limit which, roughly, is equivalent to the area under a curve on a Cartesian plane.

### Geometry, Trigonometry, Topology

Geometrical shapes such as circles, triangles, and prisms also have interesting mathematical properties. Thus, many international students will need to take classes in geometry, trigonometry, and/or topology.

Geometry is the general study of the mathematical properties of shapes, trigonometry focuses on the mathematical properties of triangles, and topology is study of the properties of continuity and contiguity.

The knowledge international students gain in algebra and calculus classes is essential for understanding the subjects in this group.

### Coursework

In answering the question “what is math?,” we can talk about the subjects that are covered within the discipline, but we can also talk about what it is to practice doing math. As you progress through courses in mathematics, you will find a change in the work expected of you.

In the earlier classes, you will often be assigned exercises and directed to solve for a solution following a mechanical set of rules. These exercises are good for developing the ability to use mathematical principles in problem-solving, and to test that one has improved on this ability.

However, as you move into more advanced classes toward your math degree, your understanding of mathematics needs to be tested more thoroughly than merely following mechanical rules. Instead of rote exercises, students will be expected to develop proofs for their homework assignments and exams. Proofs involve deriving conclusions from a set of assumptions called axioms.

Deriving proofs are often challenging and provide different set of challenges than completing exercises, but to be an expert in mathematics, one must master the ability to derive proofs.

## “What is Mathematics?” and why we should ask, where one should experience and learn that, and how to teach it

“What is Mathematics?” [with a question mark!] is the title of a famous book by Courant and Robbins, first published in 1941, which does not answer the question.

The question is, however, essential: The public image of the subject (of the science, and of the profession) is not only relevant for the support and funding it can get, but it is also crucial for the talent it manages to attract—and thus ultimately determines what mathematics can achieve, as a science, as a part of human culture, but also as a substantial component of economy and technology. In this lecture we thus

- discuss the image of mathematics (where “image” might be taken literally!),
- sketch a multi-facetted answer to the question “What is Mathematics?,”
- stress the importance of learning “What is Mathematics” in view of Klein’s “double discontinuity” in mathematics teacher education,
- present the “Panorama project” as our response to this challenge,
- stress the importance of telling stories in addition to teaching mathematics, and finally,
- suggest that the mathematics curricula at schools and at universities should correspondingly have space and time for at least three different subjects called Mathematics.

This paper is a slightly updated reprint of: Günter M. Ziegler and Andreas Loos, Learning and Teaching “What is Mathematics”, Proc. International Congress of Mathematicians, Seoul 2014, pp. 1201–1215; reprinted with kind permission by Prof. Hyungju Park, the chairman of ICM 2014 Organizing Committee.

Download conference paper PDF **Defining mathematics.** According to Wikipedia in English, in the March 2014 version, the answer to “What is Mathematics?” is

**Mathematics** is the abstract study of topics such as quantity (numbers),[2] structure,[3] space,[2] and change.[4][5][6] There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.[7][8]

Mathematicians seek out patterns (Highland & Highland, 1961, 1963) and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature.

Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist.

The research required to solve mathematical problems can take years or even centuries of sustained inquiry.

None of this is entirely wrong, but it is also not satisfactory. Let us just point out that the fact that there is no agreement about the definition of mathematics, given as part of a definition of mathematics, puts us into logical difficulties that might have made Gödel smile.1

The answer given by Wikipedia in the current German version, reads (in our translation):

**Mathematics** […] is a science that developed from the investigation of geometric figures and the computing with numbers. For mathematics, there is no commonly accepted definition; today it is usually described as a science that investigates abstract structures that it created itself by logical definitions using logic for their properties and patterns.

This is much worse, as it portrays mathematics as a subject without any contact to, or interest from, a real world.

**The borders of mathematics.** Is mathematics “stand-alone”? Could it be defined without reference to “neighboring” subjects, such as physics (which does appear in the English Wikipedia description)? Indeed, one possibility to characterize mathematics describes the borders/boundaries that separate it from its neighbors. Even humorous versions of such “distinguishing statements” such as

- “Mathematics is the part of physics where the experiments are cheap.”
- “Mathematics is the part of philosophy where (some) statements are true—without debate or discussion.”
- “Mathematics is computer science without electricity.” (So “Computer science is mathematics with electricity.”)

contain a lot of truth and possibly tell us a lot of “characteristics” of our subject. None of these is, of course, completely true or completely false, but they present opportunities for discussion.

**What we do in mathematics**.

We could also try to define mathematics by “what we do in mathematics”: This is much more diverse and much more interesting than the Wikipedia descriptions! Could/should we describe mathematics not only as a research discipline and as a subject taught and learned at school, but also as a playground for pupils, amateurs, and professionals, as a subject that presents challenges (not only for pupils, but also for professionals as well as for amateurs), as an arena for competitions, as a source of problems, small and large, including some of the hardest problems that science has to offer, at all levels from elementary school to the millennium problems (Csicsery, 2008; Ziegler, 2011)?

**What we teach in mathematics classes**. Education bureaucrats might (and probably should) believe that the question “What is Mathematics?” is answered by high school curricula. But what answers do these give?

This takes us back to the nineteenth century controversies about what mathematics should be taught at school and at the Universities. In the German version this was a fierce debate.

On the one side it saw the classical educational ideal as formulated by Wilhelm von Humboldt (who was involved in the concept for and the foundation 1806 of the Berlin University, now named Humboldt Universität, and to a certain amount shaped the modern concept of a university); here mathematics had a central role, but this was the classical “Greek” mathematics, starting from Euclid’s axiomatic development of geometry, the theory of conics, and the algebra of solving polynomial equations, not only as cultural heritage, but also as a training arena for logical thinking and problem solving. On the other side of the fight were the proponents of “Realbildung”: Realgymnasien and the technical universities that were started at that time tried to teach what was needed in commerce and industry: calculation and accounting, as well as the mathematics that could be useful for mechanical and electrical engineering—second rate education in the view of the classical German Gymnasium.

This nineteenth century debate rests on an unnatural separation into the classical, pure mathematics, and the useful, applied mathematics; a division that should have been overcome a long time ago (perhaps since the times of Archimedes), as it is unnatural as a classification tool and it is also a major obstacle to progress both in theory and in practice. Nevertheless the division into “classical” and “current” material might be useful in discussing curriculum contents—and the question for what purpose it should be taught; see our discussion in the Section “Three Times Mathematics at School?”.

**The Courant–Robbins answer**. The title of the present paper is, of course, borrowed from the famous and very successful book by Richard Courant and Herbert Robbins.

However, this title is a question—what is Courant and Robbins’ answer? Indeed, the book does not give an explicit definition of “What is Mathematics,” but the reader is supposed to get an idea from the presentation of a diverse collection of mathematical investigations.

Mathematics is much bigger and much more diverse than the picture given by the Courant–Robbins exposition. The presentation in this section was also meant to demonstrate that we need a multi-facetted picture of mathematics: One answer is not enough, we need many.

The question “What is Mathematics?” probably does not need to be answered to motivate why mathematics should be taught, as long as we agree that mathematics is important.

However, a one-sided answer to the question leads to one-sided concepts of what mathematics should be taught.

At the same time a one-dimensional picture of “What is Mathematics” will fail to motivate kids at school to do mathematics, it will fail to motivate enough pupils to study mathematics, or even to think about mathematics studies as a possible career choice, and it will fail to motivate the right students to go into mathematics studies, or into mathematics teaching. If the answer to the question “What is Mathematics”, or the implicit answer given by the public/prevailing image of the subject, is not attractive, then it will be very difficult to motivate why mathematics should be learned—and it will lead to the wrong offers and the wrong choices as to what mathematics should be learned.

Indeed, would anyone consider a science that studies “abstract” structures that it created itself (see the German Wikipedia definition quoted above) interesting? Could it be relevant? If this is what mathematics is, why would or should anyone want to study this, get into this for a career? Could it be interesting and meaningful and satisfying to teach this?

Also in view of the diversity of the students’ expectations and talents, we believe that one answer is plainly not enough. Some students might be motivated to learn mathematics because it is beautiful, because it is so logical, because it is sometimes surprising. Or because it is part of our cultural heritage.

Others might be motivated, and not deterred, by the fact that mathematics is difficult. Others might be motivated by the fact that mathematics is useful, it is needed—in everyday life, for technology and commerce, etc.

But indeed, it is not true that “the same” mathematics is needed in everyday life, for university studies, or in commerce and industry. To other students, the motivation that “it is useful” or “it is needed” will not be sufficient.

All these motivations are valid, and good—and it is also totally valid and acceptable that no single one of these possible types of arguments will reach and motivate all these students.

Why do so many pupils and students fail in mathematics, both at school and at universities? There are certainly many reasons, but we believe that motivation is a key factor. Mathematics is hard.

It is abstract (that is, most of it is not directly connected to everyday-life experiences). It is not considered worth-while.

But a lot of the insufficient motivation comes from the fact that students and their teachers do not know “What is Mathematics.”

Thus a multi-facetted image of mathematics as a coherent subject, all of whose many aspects are well connected, is important for a successful teaching of mathematics to students with diverse (possible) motivations.

This leads, in turn, to two crucial aspects, to be discussed here next: What image do students have of mathematics? And then, what should teachers answer when asked “What is Mathematics”? And where and how and when could they learn that?

## What is mathematics?

ABV-Indian Institute of Information Technology and Management Gwalior

Alumni University of Leicester & University of Sussex

University of Manitoba

KIIT University

University of Manitoba

Alumni University of Leicester & University of Sussex

KIIT University

Vivekanda college of Technology and Managment Aligarh

University of Manitoba

University of Port Harcourt

University of Manitoba

University of Manitoba

University of Manitoba

Morales Project Consulting

Ana María Sánchez Peralta

University of Granada

Ana María Sánchez Peralta

University of Granada

University of Manitoba

Universidade Federal do Paraná

University of Manitoba

University of Exeter

ABV-Indian Institute of Information Technology and Management Gwalior

ABV-Indian Institute of Information Technology and Management Gwalior

University of Manitoba

University of Manitoba

University of Manitoba

University of Manitoba

University of Manitoba

University of Manitoba

Northwestern University

University of Manitoba

Ana María Sánchez Peralta

University of Granada

University of Manitoba

University of Manitoba

University of Manitoba

Ana María Sánchez Peralta

University of Granada

University of Manitoba

Universidade Federal do Paraná

Universidade Federal do Paraná

University of Manitoba

Jamia Millia Islamia

University of Manitoba

University of Manitoba

University of Manitoba

University of Manitoba

University of Manitoba

Jamia Millia Islamia

University of Manitoba

ABV-Indian Institute of Information Technology and Management Gwalior

Alumni University of Leicester & University of Sussex

Universidade Federal do Paraná

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