mathematics as a language essay

Why Mathematics Is a Language

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Mathematics is called the language of science. Italian astronomer and physicist Galileo Galilei is attributed with the quote, " Mathematics is the language in which God has written the universe ." Most likely this quote is a summary of his statement in  Opere Il Saggiatore:

[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.

Yet, is mathematics truly a language, like English or Chinese? To answer the question, it helps to know what language is and how the vocabulary and grammar of mathematics are used to construct sentences.

Key Takeaways: Why Math is a Language

• In order to be considered a language, a system of communication must have vocabulary, grammar, syntax, and people who use and understand it.
• Mathematics meets this definition of a language. Linguists who don't consider math a language cite its use as a written rather than spoken form of communication.
• Math is a universal language. The symbols and organization to form equations are the same in every country of the world.

What Is a Language?

There are multiple definitions of " language ." A language may be a system of words or codes used within a discipline. Language may refer to a system of communication using symbols or sounds. Linguist Noam Chomsky defined language as a set of sentences constructed using a finite set of elements. Some linguists believe language should be able to represent events and abstract concepts.

Whichever definition is used, a language contains the following components:

• There must be a vocabulary of words or symbols.
• Meaning must be attached to the words or symbols.
• A language employs grammar , which is a set of rules that outline how vocabulary is used.
• A syntax organizes symbols into linear structures or propositions.
• A narrative or discourse consists of strings of syntactic propositions.
• There must be (or have been) a group of people who use and understand the symbols.

Mathematics meets all of these requirements. The symbols, their meanings, syntax, and grammar are the same throughout the world. Mathematicians, scientists, and others use math to communicate concepts. Mathematics describes itself (a field called meta-mathematics), real-world phenomena, and abstract concepts.

Vocabulary, Grammar, and Syntax in Mathematics

The vocabulary of math draws from many different alphabets and includes symbols unique to math. A mathematical equation may be stated in words to form a sentence that has a noun and a verb, just like a sentence in a spoken language. For example:

3 + 5 = 8

could be stated as "Three added to five equals eight."

Breaking this down, nouns in math include:

• Arabic numerals (0, 5, 123.7)
• Fractions (1⁄4, 5⁄9, 2 1⁄3)
• Variables (a, b, c, x, y, z)
• Expressions (3x, x 2 , 4 + x)
• Diagrams or visual elements (circle, angle, triangle, tensor, matrix)
• Infinity (∞)
• Imaginary numbers (i, -i)
• The speed of light (c)

Verbs include symbols including:

• Equalities or inequalities (=, <, >)
• Actions such as addition, subtraction, multiplication, and division (+, -, x or *, ÷ or /)
• Other operations (sin, cos, tan, sec)

If you try to perform a sentence diagram on a mathematical sentence, you'll find infinitives, conjunctions, adjectives, etc. As in other languages, the role played by a symbol depends on its context.

International Rules

Mathematics grammar and syntax, like vocabulary, are international. No matter what country you're from or what language you speak, the structure of the mathematical language is the same.

• Formulas are read from left to right.
• The Latin alphabet is used for parameters and variables. To some extent, the Greek alphabet is also used. Integers are usually drawn from i , j , k , l , m , n . Real numbers are represented by  a ,  b ,  c , α , β , γ. Complex numbers are indicated by w and z . Unknowns are x , y , z . Names of functions are usually f , g , h .
• The Greek alphabet is used to represent specific concepts. For example, λ is used to indicate wavelength and ρ means density.
• Parentheses and brackets indicate the order in which the symbols interact .
• The way functions, integrals, and derivatives are phrased is uniform.

Language as a Teaching Tool

Understanding how mathematical sentences work is helpful when teaching or learning math. Students often find numbers and symbols intimidating, so putting an equation into a familiar language makes the subject more approachable. Basically, it's like translating a foreign language into a known one.

While students typically dislike word problems, extracting the nouns, verbs, and modifiers from a spoken/written language and translating them into a mathematical equation is a valuable skill to have. Word problems improve comprehension and increase problem-solving skills.

Because mathematics is the same all over the world, math can act as a universal language. A phrase or formula has the same meaning, regardless of another language that accompanies it. In this way, math helps people learn and communicate, even if other communication barriers exist.

The Argument Against Math as a Language

Not everyone agrees that mathematics is a language. Some definitions of "language" describe it as a spoken form of communication. Mathematics is a written form of communication. While it may be easy to read a simple addition statement aloud (e.g., 1 + 1 = 2), it's much harder to read other equations aloud (e.g., Maxwell's equations). Also, the spoken statements would be rendered in the speaker's native language, not a universal tongue.

However, sign language would also be disqualified based on this criterion. Most linguists accept sign language as a true language. There are a handful of dead languages that no one alive knows how to pronounce or even read anymore.

A strong case for mathematics as a language is that modern elementary-high school curricula uses techniques from language education for teaching mathematics. Educational psychologist Paul Riccomini and colleagues wrote that students learning mathematics require "a robust vocabulary knowledge base; flexibility; fluency and proficiency with numbers, symbols, words, and diagrams; and comprehension skills."

• Ford, Alan, and F. David Peat. " The Role of Language in Science ." Foundations of Physics 18.12 (1988): 1233–42. 
• Galilei, Galileo. "'The Assayer' ('Il Saggiatore' in Italian) (Rome, 1623)." The Controversy on the Comets of 1618 . Eds. Drake, Stillman and C. D. O'Malley. Philadelphia: University of Pennsylvania Press, 1960. 
• Klima, Edward S., and Ursula Bellugi. "The Signs of Language. "Cambridge, MA: Harvard University Press, 1979. 
• Riccomini, Paul J., et al. " The Language of Mathematics: The Importance of Teaching and Learning Mathematical Vocabulary ." Reading & Writing Quarterly 31.3 (2015): 235-52. Print.
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• Definition and Examples of Text in Language Studies
• Cognitive Grammar
• Definition and Examples of Linguists
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• Generative Grammar: Definition and Examples
• 10 Test Question Terms and What They Ask Students to Do

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Mathematics: The only true universal language

By Martin Rees

11 February 2009

An alien’s description of the cosmos might teach us a thing or two about the nature of reality

(Image: Wolcott Henry/National Geographic/Getty)

IF WE ever establish contact with intelligent aliens living on a planet around a distant star, we would expect some problems communicating with them. As we are many light years away, our signals would take many years to reach them, so there would be no scope for snappy repartee. There could be an IQ gap and the aliens might be built from quite different chemistry.

Yet there would be much common ground too. They would be made of similar atoms to us. They could trace their origins back to the big bang 13.7 billion years ago, and they would share with us the universe’s future. However, the surest common culture would be mathematics.

Mathematics has been the language of science for thousands of years, and it is remarkably successful. In a famous essay, the great physicist Eugene Wigner wrote about the “unreasonable effectiveness of mathematics”. Most of us resonate with the perplexity expressed by Wigner, and also with Einstein’s dictum that “the most incomprehensible thing about the universe is that it is comprehensible”. We marvel at the fact that the universe is not anarchic – that atoms obey the same laws in distant galaxies as in the lab. The aliens would, like us, be astonished by the patterns in our shared cosmos and by the effectiveness of mathematics in describing those patterns.

Mathematics can point the way towards new discoveries in physics too. Most famously, British theorist Paul Dirac used pure mathematics to formulate an equation…

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We Should Teach Math Like It’s a Language

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The United States has a math problem, and, like most middle school students sitting down with their homework, we are not finding any easy solutions. Young people in this country are struggling to attain the proficiency necessary to pursue the careers our economy desperately needs. Universities bemoan students’ inability to complete college-level math. Each year thousands of newly admitted college students are placed in non-credit-bearing remedial courses in math, a path that immediately puts them at a higher risk of not completing a degree.

Maybe it’s the classics professor in me talking, but I approach this math problem from an unorthodox angle: Latin. In a 2011 article, “An Apology for Latin and Math,” high school Latin teacher Cheryl Lowe made a compelling comparison between the study of Latin and the study of math. Much like Latin, she observed, “math is hard because it builds so relentlessly year after year. Any skill not mastered one year will make work difficult the next.”

High school teachers have discovered that the unrelentingly cumulative nature of the study of Latin and the study of mathematics explains why students struggle to excel in either discipline.

A favorite lament of college and university faculty in quantitative fields is that students cannot perform college-level math. But what is college-level math?

In the world of classics, there is no such thing as college-level Latin. My daughter’s high school Latin teacher uses the same textbook for her class that I have used to teach Latin at Duke University, Whitman College in Washington state, and the University of Southern Maine. It turns out that there are only two differences between high school Latin and college Latin. The first is pace. I tell students that one year of college Latin is the approximate equivalent of three years of high school Latin.

The other difference is the developmental level of the student. A high school student is often not as prepared as a college student to confront demanding theoretical material, and therefore college classes might incorporate more theory than would a high school class.

The United States has been sucked into the myth of college-level math."

Like Latin, algebra is a language; and like Latin, algebra is taught to students of different developmental levels at different paces and with different levels of theoretical grounding across the K-16 landscape. The crucial difference between Latin and math education is that classicists understand that Latin is Latin, no matter the level.

Students who took Latin in high school are often encouraged to begin Latin anew when they get to college. The review students receive in an introductory college course reinforces their learning in preparation for more advanced work. This is precisely what I did when I got to college, and no one suggested that mine was not college-level study, quite the opposite: I became a classical languages major and nine years later finished a doctorate in classical studies.

In contrast, the United States has been sucked into the myth of college-level math. If students need a review of algebra, instead of encouraging them to start anew in order to reinforce their skills, we test them, label that review “remedial,” and withhold college credit from them. This message is jarring and discouraging to new college students, many of whom already have significant doubts about whether they belong in college or whether college is worth the investment.

I point out the discrepancy between our approaches to these subjects not to downplay the national crisis in quantitative reasoning but to suggest that the deficiency is in our attitude toward teaching math.

We know how to teach Latin, and we do it well. Year after year we teach the same challenging skills, facts, and concepts in different ways from middle school through college, never complaining that students are not doing college-level work. Once students have enough facility to read unabridged ancient texts, whether that happens in 8th grade or their junior year of college, we move on to translation and critical reading appropriate to the developmental level of the student.

Our approach to teaching Latin can inform better practices in math education. No one would deny that students wishing to become physicists must master calculus, but we must shift our narrative from one that labels students “deficient” on the basis of arbitrary grade-level designations. Instead, we should embrace a reverse-engineering model in which we establish clear, carefully constructed pathways to the things students must do.

Our lamentations about student deficiencies and our focus on what constitutes college-level work have been an unfortunate distraction from the salient challenge of how to help students reach the careers or paths of study to which they aspire. Meeting this challenge may require blurring the lines even further between high school and college curricula. It may require courses of different paces and configurations from the familiar K-16 standards. It will certainly require better partnerships between high schools and universities.

As is so common in the academy, we have focused on faculty-centric, content-based questions (“What constitutes college-level math?”) rather than student-centric, learning-based questions (“What do students need to reach their goals?”). Solving our math problem will require unorthodox strategies for increasing student success in math rather than trying to quantify what “counts” at the college level.

A version of this article appeared in the May 30, 2018 edition of Education Week as Math Is a Language. Let’s Teach It That Way

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Please note you do not have access to teaching notes, mathematics as a universal language: transcending cultural lines.

Journal for Multicultural Education

ISSN : 2053-535X

Article publication date: 8 August 2016

Universal language can be viewed as a conjectural or antique dialogue that is understood by a great deal, if not all, of the world’s population. In this paper, a sound argument is presented that mathematical language exudes characteristics of worldwide understanding. The purpose of this paper is to explore mathematical language as a tool that transcends cultural lines.

Design/methodology/approach

This study has used a case study approach. The data relevant to the study were collected using participant observations, video recordings of classroom interactions and field notes.

Researchers found that mathematics communication and understanding were mutual among both groups whose languages were foreign to each other. Findings from this study stand to contribute to the ongoing discussion and debates about the universality of mathematics and to influence the teaching and learning of mathematics around the world.

Originality/value

Mathematics is composed of definitions, theorems, axioms, postulates, numbers and concepts that can all generally be expressed as symbols and that have been proven to be true across many nations. Through the symbolic representation of mathematical ideas, communication may occur that stands to break cultural barriers and unite all people using one common language.

• Multicultural
• Global education
• Mathematical language

Parker Waller, P. and Flood, C.T. (2016), "Mathematics as a universal language: transcending cultural lines", Journal for Multicultural Education , Vol. 10 No. 3, pp. 294-306. https://doi.org/10.1108/JME-01-2016-0004

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Mathematics: The Language of Nature Essay

Reaction 1: The importance attached to mathematics as a science, and a collaborator for physics has fast gained credibility and recognition, especially in the context of its logical reasoning and scientific principles that supports its theories and practices. It is also seen as claiming to be the fountainhead of thoughts that have been generated through interpretations of surreal mathematics.

Reaction 2: The success gained by mathematics could be largely attributed to the fact that it is a language that has quantitative utility value; it does not relate itself with mundane business applications, but is more concerned with abstract relationships and also in terms of seeking relations within relationships, in order to critically examine the principles or theories themselves, rather than the empirical application of such tenets. (Peat & Mickens, 1990).

Reaction 3: The language of mathematics, especially in its pristine form, has been popularized because it is the only quantitative method that could possibly address theories of physics, more so, in contemporary times. The Cartesian grid, for instance, has been a striking illustration of mathematical suzerainty over physics. (Peat & Mickens, 1990).

Reaction 4: The debate whether mathematics could be treated as a language needs to be seen in the context of the fact that in certain cases, it deals with codified quantitative data which has very little to do with the abstract and conceptual thinking attributed to languages. The fact remains that mathematics may not be treated more than a technical language due to absence of conveyance of human emotions and feelings.

Reaction 5: Drawing comparison between mathematics and music, it could be said that while both follow logical order and precision, music is a sense experience that transcend normal senses, and “seeks a harmony between the four basic human functions; thought balanced by feeling and intuition by sensation.“ (Peat & Mickens, 1990).

Reaction 6: In reaching a nexus between mathematics and functioning of the human brain, it is seen that both have patterned hierarchical level of thinking and logical functioning. As a matter of fact, the brain needs to seek assimilate and correlate data in a structured and orderly manner in order to solve a mathematical problem. It is also seen that the science of mathematics also lends itself for structural integrity and coherence.

Reaction 7: The aspect of archetype cannot also be ruled out, in that scientific arguments and validations of many great mathematics have originated not from rigorous pursuit of study but from their intuitions, or gut feelings. It would also not be improbable to surmise that these hunches could form the premise of major mathematical breakthroughs in future, too. (Peat & Mickens, 1990).

Reaction 8 : It may be concluded that theories that mathematics as a precursor to physics may be valid, sustainable, and may lent credence to the reality of our very existence on earth, but it is essential that a wider perspective be taken in order to ake stock of the goals and objectives of scientific studies. It also needs to be assessed and judged in order to be able to make critical appreciation of the various empirical and scholarly treatise of mathematics as a major quantitative and value based subject amenable to interpretations and future studies.

Peat, F. David., & Mickens, Ronald E (Ed.). (1990). Mathematics and the Language of Nature . Mathematics and Sciences. (Word Scientific, 1990). (provided by the customer).

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

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• First Online: 02 November 2017
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• Günter M. Ziegler 3 &
• Andreas Loos 4  

Part of the book series: ICME-13 Monographs ((ICME13Mo))

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“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.

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What is mathematics.

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. Footnote 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.

Why Should We Care?

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?

The Image of Mathematics

A 2008 study by Mendick, Epstein, and Moreau ( 2008 ), which was based on an extensive survey among British students, was summarized as follows:

Many students and undergraduates seem to think of mathematicians as old, white, middle-class men who are obsessed with their subject, lack social skills and have no personal life outside maths. The student’s views of maths itself included narrow and inaccurate images that are often limited to numbers and basic arithmetic.

The students’ image of what mathematicians are like is very relevant and turns out to be a massive problem, as it defines possible (anti-)role models, which are crucial for any decision in the direction of “I want to be a mathematician.” If the typical mathematician is viewed as an “old, white, male, middle-class nerd,” then why should a gifted 16-year old girl come to think “that’s what I want to be when I grow up”? Mathematics as a science, and as a profession, looses (or fails to attract) a lot of talent this way! However, this is not the topic of this presentation.

On the other hand the first and the second diagnosis of the quote from Mendick et al. ( 2008 ) belong together: The mathematicians are part of “What is Mathematics”!

And indeed, looking at the second diagnosis, if for the key word “mathematics” the images that spring to mind don’t go beyond a per se meaningless “ \( a^{2} + b^{2} = c^{2} \) ” scribbled in chalk on a blackboard—then again, why should mathematics be attractive, as a subject, as a science, or as a profession?

We think that we have to look for, and work on, multi-facetted and attractive representations of mathematics by images. This could be many different, separate images, but this could also be images for “mathematics as a whole.”

Four Images for “What Is Mathematics?”

Striking pictorial representations of mathematics as a whole (as well as of other sciences!) and of their change over time can be seen on the covers of the German “Was ist was” books. The history of these books starts with the series of “How and why” Wonder books published by Grosset and Dunlop, New York, since 1961, which was to present interesting subjects (starting with “Dinosaurs,” “Weather,” and “Electricity”) to children and younger teenagers. The series was published in the US and in Great Britain in the 1960s and 1970s, but it was and is much more successful in Germany, where it was published (first in translation, then in volumes written in German) by Ragnar Tessloff since 1961. Volume 18 in the US/UK version and Volume 12 in the German version treats “Mathematics”, first published in 1963 (Highland & Highland, 1963 ), but then republished with the same title but a new author and contents in 2001 (Blum, 2001 ). While it is worthwhile to study the contents and presentation of mathematics in these volumes, we here focus on the cover illustrations (see Fig.  1 ), which for the German edition exist in four entirely different versions, the first one being an adaption of the original US cover of (Highland & Highland, 1961 ).

The four covers of “Was ist was. Band 12: Mathematik” (Highland & Highland, 1963 ; Blum, 2001 )

All four covers represent a view of “What is Mathematics” in a collage mode, where the first one represents mathematics as a mostly historical discipline (starting with the ancient Egyptians), while the others all contain a historical allusion (such as pyramids, Gauß, etc.) alongside with objects of mathematics (such as prime numbers or \( \pi \) , dices to illustrate probability, geometric shapes). One notable object is the oddly “two-colored” Möbius band on the 1983 cover, which was changed to an entirely green version in a later reprint.

One can discuss these covers with respect to their contents and their styles, and in particular in terms of attractiveness to the intended buyers/readers. What is over-emphasized? What is missing? It seems more important to us to

think of our own images/representations for “What is Mathematics”,

think about how to present a multi-facetted image of “What is Mathematics” when we teach.

Indeed, the topics on the covers of the “Was ist was” volumes of course represent interesting (?) topics and items discussed in the books. But what do they add up to? We should compare this to the image of mathematics as represented by school curricula, or by the university curricula for teacher students.

In the context of mathematics images, let us mention two substantial initiatives to collect and provide images from current mathematics research, and make them available on internet platforms, thus providing fascinating, multi-facetted images of mathematics as a whole discipline:

Guy Métivier et al.: “Image des Maths. La recherche mathématique en mots et en images” [“Images of Maths. Mathematical research in words and images”], CNRS, France, at images.math.cnrs.fr (texts in French)

Andreas D. Matt, Gert-Martin Greuel et al.: “IMAGINARY. open mathematics,” Mathematisches Forschungsinstitut Oberwolfach, at imaginary.org (texts in German, English, and Spanish).

The latter has developed from a very successful travelling exhibition of mathematics images, “IMAGINARY—through the eyes of mathematics,” originally created on occasion of and for the German national science year 2008 “Jahr der Mathematik. Alles was zählt” [“Year of Mathematics 2008. Everything that counts”], see www.jahr-der-mathematik.de , which was highly successful in communicating a current, attractive image of mathematics to the German public—where initiatives such as the IMAGINARY exhibition had a great part in the success.

Teaching “What Is Mathematics” to Teachers

More than 100 years ago, in 1908, Felix Klein analyzed the education of teachers. In the introduction to the first volume of his “Elementary Mathematics from a Higher Standpoint” he wrote (our translation):

At the beginning of his university studies, the young student is confronted with problems that do not remind him at all of what he has dealt with up to then, and of course, he forgets all these things immediately and thoroughly. When after graduation he becomes a teacher, he has to teach exactly this traditional elementary mathematics, and since he can hardly link it with his university mathematics, he soon readopts the former teaching tradition and his studies at the university become a more or less pleasant reminiscence which has no influence on his teaching (Klein, 1908 ).

This phenomenon—which Klein calls the double discontinuity —can still be observed. In effect, the teacher students “tunnel” through university: They study at university in order to get a degree, but nevertheless they afterwards teach the mathematics that they had learned in school, and possibly with the didactics they remember from their own school education. This problem observed and characterized by Klein gets even worse in a situation (which we currently observe in Germany) where there is a grave shortage of Mathematics teachers, so university students are invited to teach at high school long before graduating from university, so they have much less university education to tunnel at the time when they start to teach in school. It may also strengthen their conviction that University Mathematics is not needed in order to teach.

How to avoid the double discontinuity is, of course, a major challenge for the design of university curricula for mathematics teachers. One important aspect however, is tied to the question of “What is Mathematics?”: A very common highschool image/concept of mathematics, as represented by curricula, is that mathematics consists of the subjects presented by highschool curricula, that is, (elementary) geometry, algebra (in the form of arithmetic, and perhaps polynomials), plus perhaps elementary probability, calculus (differentiation and integration) in one variable—that’s the mathematics highschool students get to see, so they might think that this is all of it! Could their teachers present them a broader picture? The teachers after their highschool experience studied at university, where they probably took courses in calculus/analysis, linear algebra, classical algebra, plus some discrete mathematics, stochastics/probability, and/or numerical analysis/differential equations, perhaps a programming or “computer-oriented mathematics” course. Altogether they have seen a scope of university mathematics where no current research becomes visible, and where most of the contents is from the nineteenth century, at best. The ideal is, of course, that every teacher student at university has at least once experienced how “doing research on your own” feels like, but realistically this rarely happens. Indeed, teacher students would have to work and study and struggle a lot to see the fascination of mathematics on their own by doing mathematics; in reality they often do not even seriously start the tour and certainly most of them never see the “glimpse of heaven.” So even if the teacher student seriously immerges into all the mathematics on the university curriculum, he/she will not get any broader image of “What is Mathematics?”. Thus, even if he/she does not tunnel his university studies due to the double discontinuity, he/she will not come back to school with a concept that is much broader than that he/she originally gained from his/her highschool times.

Our experience is that many students (teacher students as well as classical mathematics majors) cannot name a single open problem in mathematics when graduating the university. They have no idea of what “doing mathematics” means—for example, that part of this is a struggle to find and shape the “right” concepts/definitions and in posing/developing the “right” questions and problems.

And, moreover, also the impressions and experiences from university times will get old and outdated some day: a teacher might be active at a school for several decades—while mathematics changes! Whatever is proved in mathematics does stay true, of course, and indeed standards of rigor don’t change any more as much as they did in the nineteenth century, say. However, styles of proof do change (see: computer-assisted proofs, computer-checkable proofs, etc.). Also, it would be good if a teacher could name “current research focus topics”: These do change over ten or twenty years. Moreover, the relevance of mathematics in “real life” has changed dramatically over the last thirty years.

The Panorama Project

For several years, the present authors have been working on developing a course [and eventually a book (Loos & Ziegler, 2017 )] called “Panorama der Mathematik” [“Panorama of Mathematics”]. It primarily addresses mathematics teacher students, and is trying to give them a panoramic view on mathematics: We try to teach an overview of the subject, how mathematics is done, who has been and is doing it, including a sketch of main developments over the last few centuries up to the present—altogether this is supposed to amount to a comprehensive (but not very detailed) outline of “What is Mathematics.” This, of course, turns out to be not an easy task, since it often tends to feel like reading/teaching poetry without mastering the language. However, the approach of Panorama is complementing mathematics education in an orthogonal direction to the classic university courses, as we do not teach mathematics but present (and encourage to explore ); according to the response we get from students they seem to feel themselves that this is valuable.

Our course has many different components and facets, which we here cast into questions about mathematics. All these questions (even the ones that “sound funny”) should and can be taken seriously, and answered as well as possible. For each of them, let us here just provide at most one line with key words for answers:

When did mathematics start?

Numbers and geometric figures start in stone age; the science starts with Euclid?

How large is mathematics? How many Mathematicians are there?

The Mathematics Genealogy Project had 178854 records as of 12 April 2014.

How is mathematics done, what is doing research like?

Collect (auto)biographical evidence! Recent examples: Frenkel ( 2013 ) , Villani ( 2012 ).

What does mathematics research do today? What are the Grand Challenges?

The Clay Millennium problems might serve as a starting point.

What and how many subjects and subdisciplines are there in mathematics?

See the Mathematics Subject Classification for an overview!

Why is there no “Mathematical Industry”, as there is e.g. Chemical Industry?

There is! See e.g. Telecommunications, Financial Industry, etc.

What are the “key concepts” in mathematics? Do they still “drive research”?

Numbers, shapes, dimensions, infinity, change, abstraction, …; they do.

What is mathematics “good for”?

It is a basis for understanding the world, but also for technological progress.

Where do we do mathematics in everyday life?

Not only where we compute, but also where we read maps, plan trips, etc.

Where do we see mathematics in everyday life?

There is more maths in every smart phone than anyone learns in school.

What are the greatest achievements of mathematics through history?

Make your own list!

An additional question is how to make university mathematics more “sticky” for the tunneling teacher students, how to encourage or how to force them to really connect to the subject as a science. Certainly there is no single, simple, answer for this!

Telling Stories About Mathematics

How can mathematics be made more concrete? How can we help students to connect to the subject? How can mathematics be connected to the so-called real world?

Showing applications of mathematics is a good way (and a quite beaten path). Real applications can be very difficult to teach since in most advanced, realistic situation a lot of different mathematical disciplines, theories and types of expertise have to come together. Nevertheless, applications give the opportunity to demonstrate the relevance and importance of mathematics. Here we want to emphasize the difference between teaching a topic and telling about it. To name a few concrete topics, the mathematics behind weather reports and climate modelling is extremely difficult and complex and advanced, but the “basic ideas” and simplified models can profitably be demonstrated in highschool, and made plausible in highschool level mathematical terms. Also success stories like the formula for the Google patent for PageRank (Page, 2001 ), see Langville and Meyer ( 2006 ), the race for the solution of larger and larger instances of the Travelling Salesman Problem (Cook, 2011 ), or the mathematics of chip design lend themselves to “telling the story” and “showing some of the maths” at a highschool level; these are among the topics presented in the first author’s recent book (Ziegler, 2013b ), where he takes 24 images as the starting points for telling stories—and thus developing a broader multi-facetted picture of mathematics.

Another way to bring maths in contact with non-mathematicians is the human level. Telling stories about how maths is done and by whom is a tricky way, as can be seen from the sometimes harsh reactions on www.mathoverflow.net to postings that try to excavate the truth behind anecdotes and legends. Most mathematicians see mathematics as completely independent from the persons who explored it. History of mathematics has the tendency to become gossip , as Gian-Carlo Rota once put it (Rota, 1996 ). The idea seems to be: As mathematics stands for itself, it has also to be taught that way.

This may be true for higher mathematics. However, for pupils (and therefore, also for teachers), transforming mathematicians into humans can make science more tangible, it can make research interesting as a process (and a job?), and it can be a starting/entry point for real mathematics. Therefore, stories can make mathematics more sticky. Stories cannot replace the classical approaches to teaching mathematics. But they can enhance it.

Stories are the way by which knowledge has been transferred between humans for thousands of years. (Even mathematical work can be seen as a very abstract form of storytelling from a structuralist point of view.) Why don’t we try to tell more stories about mathematics, both at university and in school—not legends, not fairy tales, but meta-information on mathematics—in order to transport mathematics itself? See (Ziegler, 2013a ) for an attempt by the first author in this direction.

By stories, we do not only mean something like biographies, but also the way of how mathematics is created or discovered: Jack Edmonds’ account (Edmonds, 1991 ) of how he found the blossom shrink algorithm is a great story about how mathematics is actually done . Think of Thomas Harriot’s problem about stacking cannon balls into a storage space and what Kepler made out of it: the genesis of a mathematical problem. Sometimes scientists even wrap their work into stories by their own: see e.g. Leslie Lamport’s Byzantine Generals (Lamport, Shostak, & Pease, 1982 ).

Telling how research is done opens another issue. At school, mathematics is traditionally taught as a closed science. Even touching open questions from research is out of question, for many good and mainly pedagogical reasons. However, this fosters the image of a perfect science where all results are available and all problems are solved—which is of course completely wrong (and moreover also a source for a faulty image of mathematics among undergraduates).

Of course, working with open questions in school is a difficult task. None of the big open questions can be solved with an elementary mathematical toolbox; many of them are not even accessible as questions. So the big fear of discouraging pupils is well justified. On the other hand, why not explore mathematics by showing how questions often pop up on the way? Posing questions in and about mathematics could lead to interesting answers—in particular to the question of “What is Mathematics, Really?”

Three Times Mathematics at School?

So, what is mathematics? With school education in mind, the first author has argued in Ziegler ( 2012 ) that we are trying cover three aspects the same time, which one should consider separately and to a certain extent also teach separately:

A collection of basic tools, part of everyone’s survival kit for modern-day life—this includes everything, but actually not much more than, what was covered by Adam Ries’ “Rechenbüchlein” [“Little Book on Computing”] first published in 1522, nearly 500 years ago;

A field of knowledge with a long history, which is a part of our culture and an art, but also a very productive basis (indeed a production factor) for all modern key technologies. This is a “story-telling” subject.

An introduction to mathematics as a science—an important, highly developed, active, huge research field.

Looking at current highschool instruction, there is still a huge emphasis on Mathematics I, with a rather mechanical instruction on arithmetic, “how to compute correctly,” and basic problem solving, plus a rather formal way of teaching Mathematics III as a preparation for possible university studies in mathematics, sciences or engineering. Mathematics II, which should provide a major component of teaching “What is Mathematics,” is largely missing. However, this part also could and must provide motivation for studying Mathematics I or III!

What Is Mathematics, Really?

There are many, and many different, valid answers to the Courant-Robbins question “What is Mathematics?”

A more philosophical one is given by Reuben Hersh’s book “What is Mathematics, Really?” Hersh ( 1997 ), and there are more psychological ones, on the working level. Classics include Jacques Hadamard’s “Essay on the Psychology of Invention in the Mathematical Field” and Henri Poincaré’s essays on methodology; a more recent approach is Devlin’s “Introduction to Mathematical Thinking” Devlin ( 2012 ), or Villani’s book ( 2012 ).

And there have been many attempts to describe mathematics in encyclopedic form over the last few centuries. Probably the most recent one is the gargantuan “Princeton Companion to Mathematics”, edited by Gowers et al. ( 2008 ), which indeed is a “Princeton Companion to Pure Mathematics.”

However, at a time where ZBMath counts more than 100,000 papers and books per year, and 29,953 submissions to the math and math-ph sections of arXiv.org in 2016, it is hopeless to give a compact and simple description of what mathematics really is, even if we had only the “current research discipline” in mind. The discussions about the classification of mathematics show how difficult it is to cut the science into slices, and it is even debatable whether there is any meaningful way to separate applied research from pure mathematics.

Probably the most diplomatic way is to acknowledge that there are “many mathematics.” Some years ago Tao ( 2007 ) gave an open list of mathematics that is/are good for different purposes—from “problem-solving mathematics” and “useful mathematics” to “definitive mathematics”, and wrote:

As the above list demonstrates, the concept of mathematical quality is a high-dimensional one, and lacks an obvious canonical total ordering. I believe this is because mathematics is itself complex and high-dimensional, and evolves in unexpected and adaptive ways; each of the above qualities represents a different way in which we as a community improve our understanding and usage of the subject.

In this sense, many answers to “What is Mathematics?” probably show as much about the persons who give the answers as they manage to characterize the subject.

According to Wikipedia , the same version, the answer to “Who is Mathematics” should be:

Mathematics , also known as Allah Mathematics , (born: Ronald Maurice Bean [1] ) is a hip hop producer and DJ for the Wu-Tang Clan and its solo and affiliate projects. This is not the mathematics we deal with here.

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Acknowledgment

The authors’ work has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 247029, the DFG Research Center Matheon, and the the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”.

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Ziegler, G.M., Loos, A. (2017). “What is Mathematics?” and why we should ask, where one should experience and learn that, and how to teach it. In: Kaiser, G. (eds) Proceedings of the 13th International Congress on Mathematical Education. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-62597-3_5

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Mathematics Is The Universal Language of Pure Logic

The progressive demand that math education be revised to promote the DEI agenda raises fundamental questions about the nature of mathematics, its relationship to society, and its role in education and life.

We already understand that the relationship between math and society is the consequence flowing from the relationship between math and science, on the one hand, and the well-known relationship between science and society, on the other hand. However, those are conclusions. We need to think about how we got there.

What is the relationship between math and science?

In his introduction to Human Diversity: The Biology of Gender, Race, and Class , Charles Murray explains,

The sciences form a hierarchy. “Physics rests on mathematics, chemistry on physics, biology on chemistry, and, in principle, the social sciences on biology,” wrote evolutionary biologist Robert Trivers. if (publir_show_ads) { document.write(\" \" + \" \"); }

This places mathematics at the foundation of all the sciences. Galileo Galilei (1564-1642), the father of modern science, was an Italian mathematician, physicist, astronomer, and natural philosopher. Frequently attributed to Galileo is the quip that “Mathematics is the language of science.”

Image by bimbimkha .

That quip is without question a true assertion, and it raises several follow-up questions, two of which are:

(1) If math is the language of science, then what is the language of mathematics?

(2) Exactly what is mathematics?

Each of these two questions has a simple but, perhaps, surprising answer.

First , The language of mathematics is ordinary modern language; that is, English, Spanish, German, Chinese, etc. This means that all that so many math-phobic people view as mathematical gobbledygook is, in reality, a compact and economical rendition of ordinary language.

For a simple example of this, the English statement “Five plus three times some real number x is twenty-three” can be rendered in math-talk as “3x+5=23”. This compact algebraic symbolism is a great convenience, facilitating the development of simple techniques for “solving the equation” that are easily within the grasp of the middle schooler.

As a more complicated example of this, consider the mathematical symbolic statement:

This is the agreed symbolism for the following statement in ordinary English:

For every positive real number epsilon there is a positive real number delta such that for every chain C from 3 to 10 with mesh less than delta and every interpolating sequence I of C, the sum formed by f, C and I is within epsilon of 20.

The average reader unfamiliar with calculus is not expected to understand that passage, but the point is that the English passage and the mathematical notation have the same meaning—and one could do the same with the mathematical notation and any other language.

And secondly , exactly what is mathematics? The answer is simple, albeit perhaps a bit mysterious: Mathematics is logic applied to a quantitative axiom system.

As mathematics is a branch of logic, it serves as a conduit for applying logic to problems. Approaching a problem mathematically is a technique for bringing logic to bear upon it.

Mathematics thus consists of only three kinds of entities: axioms, definitions, and theorems. Theorems are propositions logically deduced from the axioms and definitions.

An axiom is a proposition taken as true without proof. The axiom systems are usually (but not necessarily) deemed (assumed, believed) to describe nature, and the application of logic to the axiom is, therefore, deemed to reveal information about nature.

The key point here is that mathematics, at its essence, is logic applied to nature .

This point is subtle and not always understood, even by those who ought to understand it. In “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” an essay written by Eugene Wigner, a famous physicist, Wigner writes that “the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and there is no rational explanation for it.”

Wow! There is no rational explanation for the enormous usefulness of mathematics in the natural sciences?! Perhaps Wigner is missing something. Might a rational explanation be formulated from the fact that mathematics is logic, logic is wisdom, and “By wisdom the Lord laid the earth’s foundations; by understanding he set the heavens in place.” (Proverbs 3:19)

(See also Mario Livio, Is God A Mathematician? )

In this vein, you might also find interesting physicist Sabine Hossenfelder’s video, which marches nicely alongside her delightful book Lost in Math :

What are the axiom systems to which I refer for purposes of this essay? There are several, but the one that occupies most of college mathematics and the one with which the reader is already familiar from K-12 is the Real Number System (RNS). You have studied “properties” of the RNS in K-12 school math textbooks without being told that these “properties” are axioms of the RNS.

The axioms of the RNS fall into three classes: Eight algebraic axioms, four order axioms, and one topological axiom. You are familiar with the algebraic and order axioms of the RNS but probably not with the sole topological axiom, which is the axiom that makes calculus possible.

Sparing you the details of these 13 axioms, the point of all this has been for you to see that the arc of our interest has been from society to science to mathematics to logic.

Logic is the science of correct reasoning. Logic has two major divisions: deduction and induction. Deduction is the science of inference from premises to conclusion, and induction is the art of selecting premises for deduction.

In K-12, there is no course entitled “Logic.” Students first encounter “Intro to Logic” in college. But even though logic does not appear in K-12 expressly denominated as such, the fact that math is logic, and both math and logic are mediated by language,  means that all math work is replete through and through with logic and solving math problems, whether pure or applied. That is precisely the reason why math itself in K-12 is so critically important for success in college and in life—and why it is dangerous to America generally, and profoundly damaging to minority children specifically—to degrade mathematics in America.

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Physical Education should Always be Included in the School Curriculum

This essay about the multifaceted benefits of physical education (PE) highlights its overlooked yet vital role in school curricula globally. It emphasizes that PE extends beyond physical activity, fostering key life skills like teamwork, leadership, and resilience, and contributing to mental health and academic performance. The essay also notes PE’s role in promoting inclusion and combating public health issues like obesity. Ultimately, it argues for the importance of PE in holistic education, advocating for its enhanced presence in educational policies to support students’ overall success and well-being.

How it works

Physical education (PE) often goes unnoticed in discussions about educational policy, yet it is a staple in school curricula worldwide. While it may seem overshadowed by core academic subjects like mathematics and language arts, PE offers substantial educational benefits that go beyond mere physical activity. It serves as an important platform for holistic education, fostering crucial life skills and values.

PE is much more than just an exercise program. It is a critical environment for character development. Through participation in team sports, collaborative challenges, and individual activities, students learn vital skills such as teamwork, leadership, and resilience.

These qualities are not only applicable to the playground but also prepare students for real-world challenges.

Moreover, PE plays a significant role in promoting mental health. With rising concerns over student stress, the mental health benefits of regular physical activity are increasingly vital. Engaging in physical education helps improve mood, reduces stress, and boosts cognitive functions, supporting both academic performance and personal development.

Inclusion and diversity are central to contemporary PE programs, which strive to celebrate collective achievements and recognize individual differences. These programs often include adaptive sports, fostering an environment of acceptance and respect among students from various backgrounds.

PE is also at the forefront of addressing public health issues such as childhood obesity and sedentary lifestyles. By promoting physical activity and educating students about nutrition and healthy habits, PE lays the groundwork for lifelong health and disease prevention.

Research consistently shows that physical activity positively impacts academic performance. Students who regularly participate in physical activities tend to have better focus and memory retention, which are beneficial to their educational journey.

Finally, PE encourages a lasting appreciation for staying active. By exposing students to a variety of sports and activities, it ignites a lifelong passion for maintaining an active lifestyle, which contributes to long-term health and well-being.

In conclusion, PE is crucial for its comprehensive benefits that encompass developing key life skills, improving physical and mental health, and encouraging a healthy lifestyle. Elevating the role of PE in educational systems is essential for preparing students for success in all aspects of life, ensuring a healthier, more prosperous future.

Cite this page

Physical Education Should Always Be Included in The School Curriculum. (2024, Apr 29). Retrieved from https://papersowl.com/examples/physical-education-should-always-be-included-in-the-school-curriculum/

"Physical Education Should Always Be Included in The School Curriculum." PapersOwl.com , 29 Apr 2024, https://papersowl.com/examples/physical-education-should-always-be-included-in-the-school-curriculum/

PapersOwl.com. (2024). Physical Education Should Always Be Included in The School Curriculum . [Online]. Available at: https://papersowl.com/examples/physical-education-should-always-be-included-in-the-school-curriculum/ [Accessed: 30 Apr. 2024]

"Physical Education Should Always Be Included in The School Curriculum." PapersOwl.com, Apr 29, 2024. Accessed April 30, 2024. https://papersowl.com/examples/physical-education-should-always-be-included-in-the-school-curriculum/

"Physical Education Should Always Be Included in The School Curriculum," PapersOwl.com , 29-Apr-2024. [Online]. Available: https://papersowl.com/examples/physical-education-should-always-be-included-in-the-school-curriculum/. [Accessed: 30-Apr-2024]

PapersOwl.com. (2024). Physical Education Should Always Be Included in The School Curriculum . [Online]. Available at: https://papersowl.com/examples/physical-education-should-always-be-included-in-the-school-curriculum/ [Accessed: 30-Apr-2024]

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