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Essay on Importance of Mathematics in our Daily Life in 100, 200, and 350 words.

• Updated on  
• Dec 22, 2023

Mathematics is one of the core aspects of education. Without mathematics, several subjects would cease to exist. It’s applied in the science fields of physics, chemistry, and even biology as well. In commerce accountancy, business statistics and analytics all revolve around mathematics. But what we fail to see is that not only in the field of education but our lives also revolve around it. There is a major role that mathematics plays in our lives. Regardless of where we are, or what we are doing, mathematics is forever persistent. Let’s see how maths is there in our lives via our blog essay on importance of mathematics in our daily life. 

Table of Contents

• 1 Essay on Importance of Mathematics in our Daily life in 100 words 
• 2 Essay on Importance of Mathematics in our Daily life in 200 words
• 3 Essay on Importance of Mathematics in our Daily Life in 350 words

Essay on Importance of Mathematics in our Daily life in 100 words 

Mathematics is a powerful aspect even in our day-to-day life. If you are a cook, the measurements of spices have mathematics in them. If you are a doctor, the composition of medicines that make you provide prescription is made by mathematics. Even if you are going out for just some groceries, the scale that is used for weighing them has maths, and the quantity like ‘dozen apples’ has maths in it. No matter the task, one way or another it revolves around mathematics. Everywhere we go, whatever we do, has maths in it. We just don’t realize that. Maybe from now on, we will, as mathematics is an important aspect of our daily life.

Also Read:- Importance of Internet

Essay on Importance of Mathematics in our Daily life in 200 words

Mathematics, as a subject, is one of the most important subjects in our lives. Irrespective of the field, mathematics is essential in it. Be it physics, chemistry, accounts, etc. mathematics is there. The use of mathematics proceeds in our daily life to a major extent. It will be correct to say that it has become a vital part of us. Imagining our lives without it would be like a boat without a sail. It will be a shock to know that we constantly use mathematics even without realising the same. 

From making instalments to dialling basic phone numbers it all revolves around mathematics. 

Let’s take an example from our daily life. In the scenario of going out shopping, we take an estimate of hours. Even while buying just simple groceries, we take into account the weight of vegetables for scaling, weighing them on the scale and then counting the cash to give to the cashier. We don’t even realise it and we are already counting numbers and doing calculations. 

Without mathematics and numbers, none of this would be possible.

Hence we can say that mathematics helps us make better choices, more calculated ones throughout our day and hence make our lives simpler. 

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Also Read: How to Prepare for UPSC in 6 Months?

Essay on Importance of Mathematics in our Daily Life in 350 words

Mathematics is what we call a backbone, a backbone of science. Without it, human life would be extremely difficult to imagine. We cannot live even a single day without making use of mathematics in our daily lives. Without mathematics, human progress would come to a halt. 

Maths helps us with our finances. It helps us calculate our daily, monthly as well as yearly expenses. It teaches us how to divide and prioritise our expenses. Its knowledge is essential for investing money too. We can only invest money in property, bank schemes, the stock market, mutual funds, etc. only when we calculate the figures. Let’s take an example from the basic routine of a day. Let’s assume we have to make tea for ourselves. Without mathematics, we wouldn’t be able to calculate how many teaspoons of sugar we need, how many cups of milk and water we have to put in, etc. and if these mentioned calculations aren’t made, how would one be able to prepare tea? 

In such a way, mathematics is used to decide the portions of food, ingredients, etc. Mathematics teaches us logical reasoning and helps us develop problem-solving skills. It also improves our analytical thinking and reasoning ability. To stay in shape, mathematics helps by calculating the number of calories and keeping the account of the same. It helps us in deciding the portion of our meals. It will be impossible to think of sports without mathematics. For instance, in cricket, run economy, run rate, strike rate, overs bowled, overs left, number of wickets, bowling average, etc. are calculated. It also helps in predicting the result of the match. When we are on the road and driving, mathetics help us keep account of our speeds, the distance we have travelled, the amount of fuel left, when should we refuel our vehicles, etc. 

We can go on and on about how mathematics is involved in our daily lives. In conclusion, we can say that the universe revolves around mathematics. It encompasses everything and without it, we cannot imagine our lives. 

Also Read:- Essay on Pollution

Ans: Mathematics is a powerful aspect even in our day-to-day life. If you are a cook, the measurements of spices have mathematics in them. If you are a doctor, the composition of medicines that make you provide prescription is made by mathematics. Even if you are going out for just some groceries, the scale that is used for weighing them has maths, and the quantity like ‘dozen apples’ has maths in it. No matter the task, one way or another it revolves around mathematics. Everywhere we go, whatever we do, has maths in it. We just don’t realize that. Maybe from now on, we will, as mathematics is an important aspect of our daily life.

Ans: Mathematics, as a subject, is one of the most important subjects in our lives. Irrespective of the field, mathematics is essential in it. Be it physics, chemistry, accounts, etc. mathematics is there. The use of mathematics proceeds in our daily life to a major extent. It will be correct to say that it has become a vital part of us. Imagining our lives without it would be like a boat without a sail. It will be a shock to know that we constantly use mathematics even without realising the same.  From making instalments to dialling basic phone numbers it all revolves around mathematics. Let’s take an example from our daily life. In the scenario of going out shopping, we take an estimate of hours. Even while buying just simple groceries, we take into account the weight of vegetables for scaling, weighing them on the scale and then counting the cash to give to the cashier. We don’t even realise it and we are already counting numbers and doing calculations. Without mathematics and numbers, none of this would be possible. Hence we can say that mathematics helps us make better choices, more calculated ones throughout our day and hence make our lives simpler.  

Ans: Archimedes is considered the father of mathematics.

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Essay on Importance of Mathematics in Our Daily Life

Students are often asked to write an essay on Importance of Mathematics in Our Daily Life in their schools and colleges. And if you’re also looking for the same, we have created 100-word, 250-word, and 500-word essays on the topic.

Let’s take a look…

100 Words Essay on Importance of Mathematics in Our Daily Life

Introduction.

Mathematics is a crucial part of everyday life. It helps us make sense of the world around us and solve practical problems.

Mathematics in Daily Tasks

From shopping to cooking, we use math. It helps us calculate costs, quantities, and time.

Mathematics in Professions

In professions like engineering, computer science, and finance, math is indispensable.

Mathematics in Decision Making

Math helps us make informed decisions by analyzing data and predicting outcomes.

250 Words Essay on Importance of Mathematics in Our Daily Life

The pervasive presence of mathematics.

Mathematics, often perceived as a complex and abstract discipline, is in fact an integral part of our everyday lives. It forms the foundation for many of the decisions we make and the actions we perform daily, from managing finances to navigating directions.

A Tool for Logical Reasoning

Mathematics fosters logical reasoning and problem-solving skills. It cultivates an analytical mindset, enabling us to break down complex problems into simpler, manageable parts. This approach is not just confined to mathematical problems but extends to various real-life situations, thereby honing our decision-making abilities.

Mathematics in Technological Advancements

The rapid progress in technology, which has become an inseparable part of our lives, is deeply rooted in mathematical principles. Algorithms, which form the basis of computing, are mathematical models. The internet, smartphones, GPS, and even AI owe their existence to mathematical concepts.

Financial Management and Mathematics

Managing personal finances, a critical life skill, is essentially a mathematical exercise. Budgeting, calculating interest, understanding the implications of loans and mortgages, or even evaluating investment options, all require a good grasp of mathematics.

Mathematics and Scientific Understanding

Mathematics is the language of science. It helps us quantitatively understand and describe the physical world around us, from the trajectory of planets to the behavior of subatomic particles.

500 Words Essay on Importance of Mathematics in Our Daily Life

Mathematics, often perceived as a complex and abstract subject, is in fact deeply intertwined with our daily lives. It is the foundation of numerous activities we engage in, from basic tasks such as shopping and cooking to more complex ones like planning finances or solving problems.

The Ubiquity of Mathematics

Mathematics is everywhere. It is used in our everyday activities, often without our conscious realization. When we shop, we use mathematics to calculate prices, discounts, and taxes. When we cook, we use it to measure ingredients. When we travel, we use it to calculate distances, time, and fuel consumption. Even in our leisure activities such as playing games or music, mathematics plays a crucial role in understanding patterns, probabilities, and rhythms.

Mathematics in the Professional Sphere

Mathematics and problem-solving.

Mathematics also enhances our problem-solving skills. It teaches us to approach problems logically and systematically. It encourages us to break down complex problems into simpler parts, solve them individually, and combine the solutions to solve the original problem. This skill is not just applicable to mathematical problems but to any problem we encounter in life.

Mathematics and Critical Thinking

Furthermore, mathematics fosters critical thinking. It trains us to question assumptions, identify patterns, and draw conclusions based on evidence. It also teaches us to understand the limitations of our solutions and consider alternative approaches. These are valuable skills that can be applied in various aspects of life, from making informed decisions to evaluating the credibility of information.

Mathematics and the Digital Age

In conclusion, mathematics is not just a subject we learn in school. It is a powerful tool that helps us understand and navigate the world around us. It enhances our problem-solving and critical thinking skills, and it opens up a world of opportunities in the professional sphere. Therefore, it is essential that we appreciate the importance of mathematics in our daily lives, and strive to improve our mathematical literacy.

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What is the importance of mathematics in our daily lives?

Mathematics is a powerful tool for global understanding and communication that organizes our lives and prevents chaos. Mathematics helps us understand the world and provides an effective way of building mental discipline.

Math encourages logical reasoning, critical thinking, creative thinking, abstract or spatial thinking, problem-solving ability, and even effective communication skills. Let's understand the importance of mathematics in our daily life.

The Importance of Mathematics in Our Daily Lives

Mathematics : introduction.

• Mathematics helps to develop the ability to think.
• It helps explain how things work.
• It helps to develop wisdom.
• It increases the speed of intuition.
• It helps to make the child smarter.
• Money can be collected in mathematics when used as a profession.
• It is important in a constantly evolving world.
• It provides the child with an opportunity to get to the world.

Mathematics in Our Everyday Life

Although the importance of Mathematics can never be denied, a general fear of dealing with math exists in students across the world.

Having said that, most people, nowadays grapple with the calculations, as they find them too tough to handle. 

To ease their life, there exists a comprehensive platform like Khanacademy and mathisfun. Moving from specific to general, it has a host of calculators dealing with physics, chemistry, general arithmetic’s, and many more.

 So, students of various disciplines can use this website to solve their math’s-related problems without any hassle.

The Most Important Uses of Mathematics 

Practical uses of mathematics in everyday life, the importance of mathematics to individuals, the importance of mathematics to society, نموذج الاتصال.

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Use of Mathematics in Daily Life Essay | Essay on Use of Mathematics in Daily Life for Students and Children in English

February 13, 2024 by Prasanna

Use of Mathematics in Daily Life Essay:  Math has become an inseparable piece of our lives. Whether we work in an office or invest the majority of our energy at home, every last one of us utilizes math as a piece of our regular day to day existence.

Regardless of where we are, just as we are doing, math is consistently there if you notice it. For those of you who feel restless about utilizing mathematics, here is a rundown of the day by day exercises that include utilizing this general subject. So kick back and investigate how we use math in our regular day to day existence.

You can also find more  Essay Writing  articles on events, persons, sports, technology and many more.

Long and Short Essays on Use of Mathematics in Daily Life for Students and Kids in English

We provide the student with essay samples on an extended essay of 500 words and a short piece of 150 words on the Use Of Mathematics In Daily Life.

Long Essay on Use of Mathematics in Daily Life 500 Words in English

Long Essay on Use of Mathematics in Daily Life is usually given to classes 7, 8, 9, and 10.

Mathematics is a deliberate utilization of issue. Moreover, the subject of Mathematics is one of the main subjects of our life. Regardless of which field or calling you to have a place with, its utilization is all over. That is the reason it is essential to have a decent comprehension of the subject. Even though the essentials of mathematics start from school; however, its use proceeds till we become grown-ups, and in this manner, it very well may be said that maths has become a vital part. Envisioning our lives without it resembles a boat without a sail. You may be shocked to realize that we use mathematics consistently, even without knowing it. From dialing numbers on the telephone to giving cash for making the installments, our reality is encircled by mathematics. So let us see what sway this subject has made on our lives.

While planning food, we generally measure the various fixings to cook the ideal amount as it were. This is conceivable only because of mathematics. On the off chance that we didn’t know about the numbers, it would not have been reasonable to quantify, make changes, and cook delicious food. So you can say that our day starts with the idea of maths.

While going shopping, we set up a rundown of things we require, compute the necessary measure of cash for it, and so forth. This count depends on numbers which come from mathematics. Without the numbers, you can’t choose the amount you have to pay to the merchant and the amount you have spared. So on account of shopping as well, you are encircled by the universe of mathematics.

This is where various mathematics ideas are applied, and accordingly, the specialists need to have a decent agreement and order of the subject. Taking care of the bank’s exchanges isn’t straightforward, and you have to have some information on mathematics to keep up your record, store and pull out cash, and so forth. On the off chance you wish to advance, you have to consider the interest you should pay and the month to month premium you would need to pay. The financial area is identified with maths; thus, even the clients should be acquainted with it.

Everybody loves to travel, yet there is much more to it than the delight. While arranging your excursion, you do not just need to choose where you wish to go yet book your inn, tickets, and so forth. This requires spending arranging and a feeling of comprehension of mathematics with the goal that you can achieve the various assignments effectively. From making a trip separation to its cost, transport tickets, employing taxis, and so forth all require maths.

Subsequently, from the above models, you may have a reasonable thought that there is no such region where the idea of mathematics isn’t utilized. You can’t manage without this subject, and that is why it is essential to keep your rudiments option to play out the regular exercises of life. So mess around with the number and appreciate.

Short Essay on Use of Mathematics in Daily Life 150 Words in English

Short Essay on Use of Mathematics in Daily Life is usually given to classes 1, 2, 3, 4, 5, and 6.

Mathematics is helpful in our everyday life. The utilization of mathematics causes us to play out various assignments without breaking a sweat. In this article, we will find out about a couple of employments of mathematics today by day life. One of the regular employments of mathematics is cash the executives. These abilities will assist you in overseeing money. With the assistance of mathematics abilities, they can ascertain straightforward and accruing funds without much of a stretch.

Moreover, you can now access more Essay Writing on this topic and many more. These aptitudes are significant in overseeing cash and developing it up. Mathematics helps individuals to figure which bank is giving better revenue on their venture for a particular period. It will help with choosing which Mastercard and credits charge a low pace of interest. These are a portion of the significant territories in the day by day life that requires numerical aptitudes. Information on math helps you make better choices throughout everyday life, which helps make life simpler.

10 Lines on Use of Mathematics in Daily Life in English

• Mathematics is helpful in our everyday life.
• Mathematics is a deliberate utilization of issue.
• Mathematics is one of the main subjects of our life.
• Information on math helps you make better choices throughout everyday life, which helps make life simpler.
• The financial area is identified with maths; thus, even the clients should be acquainted with it.
• Mathematics helps individuals to figure which bank is giving better revenue on their venture for a particular period.
• Math has become an inseparable piece of our lives. Whether we work in an office or invest the majority of our energy at home, every last one of us utilizes math as a piece of our regular day to day existence.
• With the assistance of mathematics abilities, they can ascertain straightforward and accruing funds without much of a stretch.
• From dialing numbers on the telephone to giving cash for making the installments, our reality is encircled by mathematics.
• The universe of mathematics encompasses you.

FAQ’s on Use of Mathematics in Daily Life Essay

Question 1. What are the employments of maths?

Answer: Mathematics makes our life deliberate and forestalls tumult. Specific characteristics supported by mathematics are the intensity of thinking, innovativeness, theoretical or spatial reasoning, basic reasoning, critical thinking capacity, and even viable relational abilities.

Question 2. How would we use math in shopping?

Answer: Math utilized when purchasing something. To start with, you use including and expansion used to tell how much cash you have. Rates and development are being used to figure the price tag. At that point, the deduction is used to sort out how much change is expected.

Question 3. How would you use math when voyaging?

Answer: Math proves to be useful when voyaging. Consider it: When you travel, math tags along – from assessing the fuel measure, you’ll have to arrange out an excursion dependent on miles every hour and separation voyaged. Computing fuel use is pivotal to significant distance travel.

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Home — Essay Samples — Science — Mathematics in Everyday Life — Mathematics In Everyday Life: Most Vital Discipline

Mathematics in Everyday Life: Most Vital Discipline

• Categories: Mathematics in Everyday Life

About this sample

Words: 795 |

Published: Mar 14, 2019

Words: 795 | Pages: 2 | 4 min read

Works Cited

• Benacerraf, P. (1991). Mathematics as an object of knowledge. In P. Benacerraf & H. Putnam (Eds.), Philosophy of mathematics: Selected readings (pp. 1-13). Cambridge University Press.
• EdReady. (n.d.). Home. Retrieved from https://www.edready.org/
• Puttaswamy, T. K. (2012). Engineering mathematics. Dorling Kindersley (India) Pvt. Ltd.
• Steen, L. A. (Ed.). (2001). Mathematics today: Twelve informal essays. Springer Science & Business Media.
• Suter, B. W. (2012). Mathematics education: A critical introduction. Bloomsbury Academic.
• Tucker, A. W. (2006). Applied combinatorics. John Wiley & Sons.
• Vakil, R. (2017). A mathematical mosaic: Patterns & problem solving. Princeton University Press.
• Wolfram MathWorld. (n.d.). MathWorld--The web's most extensive mathematics resource. Retrieved from http://mathworld.wolfram.com/
• Wu, H. H. (2011). The mis-education of mathematics teachers. Educational Studies in Mathematics, 77(1), 1-20.
• Ziegler, G. M., & Aigner, M. (2012). Proofs from THE BOOK. Springer Science & Business Media.

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Why Math Is Important: A Student’s View (Updated for 2024)

A couple of weeks ago, I asked my son to write an extra essay for a project we were working on for the Classical Conversations Practicum . I allowed him to work on that essay instead of his math lesson for the day.

Suddenly, my daughter, Ada, did not want to do her math lesson for the day. I explained that my son was writing an essay instead and she asked to be allowed to do the same.

I thought, “Hmm… This is a good time for my daughter to think about why math is important and come up with her own reasons for studying this subject.” She wrote the following essay and we thought it might help parents and students to read Ada’s thoughts.

Pure mathematics is, in its way, the poetry of logical ideas.” – Albert Einstein

“Why Is Math Important?” by Ada Bianco

“Everyone agrees that learning math can be difficult, but some people believe math is important and some people believe math is not important. Math is important for three reasons:

• Math is everywhere.
• Children need math.
• God created math.

Math Is Everywhere

The first reason math is important is because it is everywhere. It is used in everyday life. It is useful, but it is more than just useful. Math is there to help us, to keep us well ordered, to help us learn new things and to help teach us new things. Students will become adults who will use math in their jobs. All kinds of careers use math. For example, musicians, accountants, fashion designers, and mothers use math. However, math is not only used for things you do. It also brings order to everything around you. The world is organized essentially because it was made with math.

Children Need Math

The second reason math is important is children need math. Now, as we all know, children are as chaotic as a volcanic eruption. But as they grow, children need to learn patience. Patience is precisely what math teaches us. It also teaches us curiosity. For example, why is this rule used here? Why would that number be negative? Why is that equation set up like that? These are questions they will learn to ask if they are taught math. The parents’ job is to help their children grow up to become good people who are patient and wise, who want to learn even more about anything and everything. Their future depends on what they have learned and if they have learned mathematics, then they will be able to do many different things—maybe even anything—when they are adults

God Created Math

The third reason math is important is God created it. This is a reason most adults use to convince their children that math is not boring and unimportant, so it may seem unoriginal. I believe, however, it is something that needs to be stated. God created the universe as well as math. The universe is full of math and it is orderly because of math. The sun is a certain distance from the earth; everything is organized in such a way that no matter what has happened we have always been safe. We need math. From this, you should be able to see how much we really do use and need math. We would not be able to process or even do everyday things without it. Math, in addition to these things, helps us to know God. God gave us math to live well and to serve Him. With everything we learn using math in science, we learn more about the world, which can help draw us closer to God.

Some people say math is unimportant because you don’t need math other than basic math principles—you can live without more complicated math. They say, if you need it, then simply use a calculator and leave the more complicated math to people who like math, the mathematicians. This, however, is not correct. You need math and could not live well without math, even including more complicated math concepts. God made us with a sense of curiosity so we can learn, do, and think about all sorts of things. Math is that thing that connects everything together, everything people love to do: music, cooking, painting, and everything else. Math is important.

Math is important because math is everywhere, children need math, and God created math. This matters to me and other children because math determines our future and how we choose to live.”

Understanding Why We Learn Subjects

Often, it’s not just our students who struggle to understand why math is important. Some of us can use a reminder as well. Hopefully Ada’s essay inspired you with reasons why we and our students should learn math. What students learn from math, just as with studying any other subject, can be applied to all areas of life. Having your student write a persuasive essay like the one above can be a great way to help them understand how a subject they are studying is useful in everyday life.

Of course, your student’s topic doesn’t have to be math. It can be history, geography, English, literature, Latin, science, or any other subject. Whatever subject they write about, this exercise is undeniably useful for helping them to understand the point of studying that subject. Perhaps they’ll even surprise you with reasons you haven’t thought of.

-Ada Bianco

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Why is Math Important: Benefits of Learning Math at School

Updated on April 29, 2024

Why is math important is a question worth exploring. Mathematics, a subject beyond mere numbers and formulas, constitutes the core of our existence. Its influence extends far beyond the confines of textbooks, penetrating the very essence of modern life. The topic — why is math the most important subject — also carries weight within the realm of education, which is why kids may be asked to write a why is math important essay in class. 

As we embrace math in education, we enable ourselves to unravel the mysteries of our reality. Through this article, you will discover answers to the question, why is math so important, and understand the many benefits of immersing ourselves in mathematics.

Why Is Math Important for Kids to Learn?

Math plays a significant role in everyone’s educational journey, bringing many benefits beyond just numbers. From the basics like counting and recognizing shapes to more complicated aspects like algebra, geometry, and calculus, studying math grounds students intellectually. At its heart, math teaches discipline and accuracy.

As people study math, they learn to take logical steps, follow the rules, and pay attention to the details. These skills make their studies easier and help them in other areas of life, teaching them how to approach problems systematically. Math also hones critical thinking and analysis.

It’s essential to know the answer to the question — why is math important for kids. When faced with math problems, we learn to spot patterns, make connections, and develop hypotheses. This natural problem-solving pathway helps us understand how things work and resolve complex issues. Besides, math literacy is a must-have in a world full of data and tech. Knowing the ins and outs of math gives kids the ability to interpret numbers and make well-thought-out decisions in terms of finance, health, and science. 

Also, math takes students and even teachers to the apex of creativity! When both parties explore numbers, shapes, and equations, they use their imaginations and develop new ways to solve problems and develop ideas. Finally, math encourages collaboration. Group activities and conversations about math help them communicate better, learn together, and make friends. 

Having understood the overview of math’s relevance in people’s lives, let’s delve deeper into why is math important in everyday life for kids.

Math Hones Complex Problem-Solving Skills

Knowing the answer to the ‘why is math important in life’ question enables kids to break down complex problems into smaller components, identify pertinent variables, and use appropriate formulas or methods to arrive at practical solutions. Math equips kids with a structured approach to problem-solving, empowering them to overcome obstacles and adapt to a dynamic world.

The capacity to methodically resolve issues enables them to approach various challenges with unwavering confidence and creativity, whether resolving complex technical troubleshooting issues, streamlining workflows, or interpersonal conflicts. 

It Promotes Critical Thinking

Knowing 5 reasons why math is important reveals math’s role in fostering critical thinking. The journey of solving mathematical problems is crucible for developing critical thinking. As kids immerse themselves in scrutinizing data, solving maze-like word problems, and developing logical strategies, they develop a robust skill — evaluating information from diverse perspectives. 

This ability to see recurring patterns and coherent conclusions is essential to making informed decisions. In debate and dialogue, kids with sharp critical thinking demonstrate the ability to obtain reliable sources, deconstruct complex arguments, and participate meaningfully in discussions. When faced with unexpected circumstances or a whirlwind of rapidly changing scenarios, this honed analytical skill allows them to objectively weigh new information, seamlessly adjust strategies, and deftly navigate the tides of change.

Math Improves Kids Financial Knowledge

Why math is the most important subject is validated within academics, but we can look beyond that. Math knowledge is a vital component of financial literacy, as it provides kids with the understanding and tools to make informed decisions that shape their financial well-being. Mathematics is central in helping individuals develop the essential skills to decipher complex financial concepts.

From understanding the dynamics of interest rates and the complex effects of investing to evaluating risk and return profiles, mathematics provides the basis for building a solid financial foundation. Using these mathematical insights, kids can create an adequate budget that meets their goals and desires. But financial literacy goes beyond self-interest; it enables them to contribute positively to their communities. 

By making intelligent philanthropic decisions or supporting local businesses, financially savvy kids become agents of change that drive economic growth and community development. Financial literacy provides clarity about a student’s perspective on broader financial issues. 

Using mathematical reasoning, they can engage in informed discussions about public policy, evaluate economic proposals, and make informed choices with far-reaching societies. The combination of mathematics and financial literacy, allowing them to secure their financial future and actively participate in creating a more financially stable and fair society, makes us more confident in answering the question, why is math important?

It Helps Kids Develop Technical Skills

In a digital age where technology permeates every aspect of modern life, the question of ‘why is discrete math important’ is quickly answered. Look at cybersecurity, for example. In 2023, it is among the most sought-after technical skills as companies try to protect their networks and data from breaches and uphold customers’ privacy.

A good understanding of mathematics opens up the ability to understand, analyze and innovate in a complex digital environment. Knowledge of mathematics allows kids to contribute to the development of technology actively. 

As technology evolves and shapes the future, mathematicians are uniquely positioned to drive progress. Using mathematical principles, they confidently explore the digital world, contributing to developing new solutions, advanced applications, and transformative breakthroughs that move society into uncharted territories of technological innovation.

Math Opens The Door to More Career Opportunities

Kids know why math is important and impacts job opportunities because of how many more career paths it offers them.

Beyond the bounds of traditional math-oriented roles like engineering and finance, the need for math skills has permeated many industries. Meanwhile, dynamic marketing has used statistical analysis to discover consumer behavior, improve customer segmentation, and drive strategic campaigns. 

As artificial intelligence and automation redefine industries, kids with a solid foundation in mathematics have the adaptability and innovation to thrive in new areas of employment in the future. From harnessing the power of big data to building data-driven narratives, these math-savvy professionals are at the forefront of shaping the future of work.

Learning Math Improves Analytical Skills

Mathematical analysis is crucial for developing analytical thinking, an invaluable skill in our complex, information-saturated world. So why is it important to learn math to improve analytical skills? In an age where navigating massive data sets and deciphering multifaceted challenges is the norm, the ability to discern complex situations and evaluate evidence becomes valuable. 

In a world where career paths and problem-solving paradigms are evolving at an unprecedented rate, the enrichment provided by mathematical and analytical ability is a cornerstone of success. Whether driving an industry into the future or developing innovative solutions to global problems, kids with these skills are built to make a lasting and transformative impact.

Progressive Scientific Discovery

The question — why is math and science important — is a run-off of the belief that math is often the language of science. Math is an indispensable tool for pushing the boundaries of scientific research and inquiry. It is the hidden force behind the breakthrough discoveries that allow scientists to bridge the gap between theoretical concepts and empirical observations.

Clinical trial design and medical analysis are governed by mathematical principles, which aid researchers in evaluating the efficacy of interventions and treatments. Statistical methods rooted in mathematics can provide insight into the effects of new drugs, the spread of diseases, and the impact of public health initiatives. This quantitative approach that improves medical knowledge and saves lives by guiding evidence-based medical practice shows why learning math is important.

It Helps Kids Develop Mental Stamina and Endurance

Facing complex math problems develops mathematical and endurance skills. It promotes mental strength and a will to overcome difficulties. When kids become aware of this, before you point out 5 reasons why math is not important they can already give you countless reasons why math is important. That is because they have gone through the rigors of solving math and now understand that mastery requires dedication, persistence, and a willingness to face failure. 

They develop an inherent resilience beyond mathematics as they solve complex problems and grapple with confusing concepts. This little thing becomes the foundation of personal and professional success. Those who successfully navigate the difficulties of mathematics are better prepared to face the complexity of the modern world. 

As we ponder why math is important in life, we should know that math provides a compass for navigating complexity in a world of information and rapid developments. This article could have still gone ahead to give an extra 10 reasons why math is important as its importance is countless. But you get the point already!

Mastering mathematics nurtures critical thinking, problem-solving abilities, and analytical reasoning — qualities necessary in a world filled with complex challenges and diverse opportunities. These all make us understand why math is important in our daily lives.

But why is learning math important at Brighterly? Brighterly recognizes the transformative power of mathematics and its role in shaping resilient individuals. They provided a platform that supports math understanding and learning. So register now to embark on a journey of discovery, where interactive lessons, engaging activities, and a supportive community await.

Jessica is a a seasoned math tutor with over a decade of experience in the field. With a BSc and Master’s degree in Mathematics, she enjoys nurturing math geniuses, regardless of their age, grade, and skills. Apart from tutoring, Jessica blogs at Brighterly. She also has experience in child psychology, homeschooling and curriculum consultation for schools and EdTech websites.

Are you seeking to transform math into something captivating and enjoyable for your students? This article is just the thing you need! It contains fascinating and fun math trivia questions and answers designed to fascinate kids. Whether you are a teacher or parent, rest assured the trivia questions math presented in this article will benefit […]

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Essay Samples on Mathematics in Everyday Life

Math: the efficient and effective methods to study math.

Math is everywhere; it should be one of the wonders of the world. In a way, Math is a fundamental part of who I am. It’s always been there for me. Yes, a bit strange coming from a high school student. Usually students despise quadratic...

• Mathematics in Everyday Life

Math Discovery and Mathematical Patterns in Standard of Living

Mathematics is literally defined as the study of numbers, quantities, formulas and patterns but in my own understanding, it is the world of numbers and with that it is how the world works. Mathematics is also the study of things, the relationships between things, and...

Problem Solving: Use of Math in Our Everyday Life

What I say about math is that I really don’t like it, but at the end of the day through high school math I have learned how to solve problems and not give up when I don’t fully understand something. I dislike math, but I do need it. The reason why I dislike math is that...

• Problem Solving

Doubt as a Key to Mathematical Knowledge

In my arabic culture, doubt, especially when directed at supperiors, is considered extremely disrespectful. In contrast the proverb, “Doubt is the key to knowledge” indicates that doubt should be looked at in a positive light and specifically as a way of knowing. However doubt is...

The Application of Persistence and Perseverance in Mathematics

All children can benefit from studying and developing strong skills in mathematics. Primarily, learning mathematics improves problem-solving skills, and working through problems can teach persistence and perseverance. Mathematics is essential in daily life for such activities as counting, cooking, managing money, and building things. Beyond...

• Persistence

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The Essence of Mathematics and Its Significance Towards the Behavior of Nature

Mathematics plays an integral part in our daily living, because everything we see, touch, and feel you can’t hide the fact that there is a math involved. Earth doesn’t appear how it looks today if scientists, don’t compute or used mathematics in making our world...

• Mathematical Predictions

The Power of Mathematics: Unveiling its Influence on Nature and Phenomena

Introduction Mathematics plays an integral part in our daily living, because everything we see, touch, and feel you can’t hide the fact that there is a math involved. Earth doesn’t appear how it looks today if scientists, don’t compute or used mathematics in making our...

• Role of Education

The Meaning of Application of Principles in Real Life

The realm of mathematics have a variety of implications on many real word activities that take place in today’s society. From the construction of a buildings to the usage of models in stocks and investment, math has a very effective role in the productivity and...

• Life Without Principle

The Relationship Between Mandala and Mathematic Studies

It is an undeniable fact that numbers have an impact on our lives and cover a very large part of our lives. Although many people think that mathematics consists of only symbols and specific rules, in spite of it seems complex when you look into...

Discovering the Effectiveness of College Algebra

Mathematics education at the college level is facing many challenges. These challenges are occurring at a time when most experts believe that students are going to need stronger mathematical skills than ever before in order to compete in the workforce (National Council of Teachers of...

• College Students

A Report On The Fibonacci Sequence

“Number rules the universe” ~ Pythagoras Numbers are found everywhere in in all aspects of human life. From the start of the day till the moment we fall asleep, we are surrounded with technology of which lots are made possible by numbers. One example are...

• Mathematical Models

The Main Drivers of My Fascination of Mathematics

One of the most striking aspects of mathematics for me is how something so seemingly abstract can have such a major purpose in the inner mechanisms of the universe. For example, which number, when multiplied by itself, is -1? By inspection, you can see that...

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• Personal Life

The Use Of Probability Theorem In Everyday Life

Throughout daily life probabilty usage is prevalent throughout all hours of life. Barometers can't anticipate precisely how climates manifest, but utilizing apparatuses and special equipment to decide probability for certain types of weather. By example if there's a certain possibility for drizzle, at that point...

Mathematics Is Not Scary, It’s Beautiful

Mathematics is often times seen as dark and scary. People specially students tried to avoid it. There are students are students tries to take courses that doesn’t have math. Well in fact it is unavoidable since math is seen everywhere and it is not scary....

Beauty Is The Creation Of Mathematics

“You are beautiful no matter what they say words can't bring you down, Oh no, You are beautiful in every single way yes words can't bring you down, Oh no, so don't you bring me down today”, sang by Christina Aguilera. I love this song...

The Beautiful Nature Of Mathematics

Beauty, as its definition given by Miriam Dictionary, is the quality or aggregate of qualities in a person or thing that gives pleasure to the senses or pleasurably exalts in the mind or spirit. In addition, as Cambridge Dictionary, is the quality of being pleasing,...

• Golden Mean

Mathematics In Every Aspect Around Us

Mathematics, as complex and absurd it may sound, is literally everywhere. Everywhere in a sense that it is frequently applied in our day-to-day activities, such as cooking (when we make correct measurements of ingredients), planning our daily agenda (how much time we will allocate for...

Mathematics Is Not Just About Numbers, It’s Also About Beauty

It has been said that “Beauty is in the eye of the beholder” for which many may believe is correct, but for some individuals does not accept this quotation. Yes, many may have their differences when it comes to preferences of foods, clothing, gadgets and...

Perfection And Beauty: My Vision Of Mathematics

How we can say Mathematics is perfect? For me Math is perfect because of the many uses or the different uses of mathematics, the benefits of math and the involvement of math in our daily life and how math contributes in our daily life. Mathematics...

The Role Of Mathematics In Creating Beauty

The beauty of mathematics is that it is not a mathematical equation, but rather a concept. I am one of those who believes that mathematics fits not only human life but also in the beauty of nature. It incorporates as part and forms everything in...

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Best topics on Mathematics in Everyday Life

1. Math: The Efficient and Effective Methods to Study Math

2. Math Discovery and Mathematical Patterns in Standard of Living

3. Problem Solving: Use of Math in Our Everyday Life

4. Doubt as a Key to Mathematical Knowledge

5. The Application of Persistence and Perseverance in Mathematics

6. The Essence of Mathematics and Its Significance Towards the Behavior of Nature

7. The Power of Mathematics: Unveiling its Influence on Nature and Phenomena

8. The Meaning of Application of Principles in Real Life

9. The Relationship Between Mandala and Mathematic Studies

10. Discovering the Effectiveness of College Algebra

11. A Report On The Fibonacci Sequence

12. The Main Drivers of My Fascination of Mathematics

13. The Use Of Probability Theorem In Everyday Life

14. Mathematics Is Not Scary, It’s Beautiful

15. Beauty Is The Creation Of Mathematics

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The Importance of Mathematics in Everyday Life

How it works

Math, as we know it, is a discipline that everyone needs in our day-to-day lives. Without it, the world would be a dangerous place to live. Would it even be feasible to implement most of the presently achieved inventions and discoveries? The answer would simply be no! Math has played a significant role in shaping the world into its current state of progress (Benacerraf, P. 1991.) As fascinating as it sounds, it has proven to be challenging for some individuals, yet this largely depends on the individual (Benacerraf, P.

To begin with, I would like to profoundly state how I have been positively influenced towards developing an interest in Math. Numbers have always created a conducive environment for me since I started learning. When I embarked on my primary education, the first thing that fascinated me was learning how to count. It was intriguing knowing that I could count like any other adult who taught me how to do so. Not to mention, my first math exam was a great success. Being the best in my math class made me fall deeply in love with math. I could not get it off my mind. Whenever it comes to this discipline, I have always had a strong sense of belonging. This is largely because I have always found it to be unique in comparison to most subjects, and I have always performed well from the very time I was introduced to it. Having always been quite confident, I know in my heart that I only need to enhance my quick thinking skills.

Secondly, as I pursue my major in Telecommunications Engineering, it’s crucial to note the units I will undertake in Math, such as Analytical Geometry and Calculus (Puttaswamy, T. K. 2012.) These two units will play an essential role in ensuring I complete my major and progressively move towards success. Analytical Geometry is vital to gain a deeper understanding, especially in Trigonometry, as it will influence the sine and cosine types of waves studied in the major. Moreover, calculus will greatly assist me in understanding how to solve equations given different types of expressions (Stillwell, J. 2010.) In my career, Math will be incredibly valuable as it will ensure precision during lengthy mathematical computations and operations. Furthermore, it will be used to approach various problems from different perspectives without limiting oneself to a specific method. Apart from my studies and career, Math also plays an indispensable role in my daily life, including carrying out simple calculations accurately, future financial planning, and basic counting that can be applied at home.

Last but not least, I made use of a net that provided terrific help when dealing with my mathematical skills. The website, known as EdReady, has proven to be quite beneficial and a potential solution to anyone’s mathematical problems. The website presented several study topics in a well-explained form, and the brilliance was thoroughly illustrated through examples with detailed guidance. As a result, even the least knowledgeable student would be able to gain some understanding. In addition to this, study questions were provided for further practice even when offline. This turned out to be quite useful since it taught me how to understand when to think out of the box or follow a straightforward approach based on the demands of a question. As this was one of my weaknesses, I was finally able to put an end to this unnerving fear, and now I can breathe easily with a clear mind. Not to mention, the website was quite easy to navigate from the very start, guiding one to register for an account and understanding how to tackle the lessons.

In conclusion, I confidently state that Math is a crucial discipline in everyone’s life. It enables us to have a flexible thought process for solving problems as there can be several different approaches in mathematics to tackle a single problem. It also sharpens one’s analytical acumen to avoid unnecessary mistakes and aim for accuracy. Honestly, Mathematics could be the best thing that has ever happened to anyone. Taking one step at a time, it does not take forever to achieve success in Math. Without this discipline, the world would definitely be deprived of many benefits and wouldn’t be as pleasant a place to live.

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September 2, 2017

Why Is It Important to Study Math?

What’s the point of learning math? Why is it so important that kids are exposed to mathematical thinking? And what do parents and teachers need to know about learning  real  math? Keep on reading to find out.

By Math Dude Jason Marshall

PeopleImages  Getty Images

Today is a very special episode of the Math Dude. To begin with, it’s episode 300. And because we humans have 10 fingers, we love to give special meaning to multiples of 10. But while that’s fun, it’s not the big news of the day or what makes this episode special to me. The big news is that this 300th episode is my last. Between my day job as a physics and astronomy professor and my day-and-night job of being “Dad” to an awesome and bustling 3-year-old, my free time for Math Dude duties has dwindled. And although I will surely miss all of you math fans, after seven years on the job, it's time to say goodbye.

But before I go, I have one more thing to say—and I think it’s the most important thing I’ve ever said on the show. It’s not something that I would (or even could) have said when I wrote the first episode seven years ago, because I wasn’t yet a father and so I wasn’t yet watching somebody discover the world for the first time. So please take a few minutes and listen, because I think this is something that everybody who has kids or might have kids or works with kids or might work with kids should know.

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What Students Are Saying About the Value of Math

We asked teenagers: Do you see the point in learning math? The answer from many was “yes.”

By The Learning Network

“Mathematics, I now see, is important because it expands the world,” Alec Wilkinson writes in a recent guest essay . “It is a point of entry into larger concerns. It teaches reverence. It insists one be receptive to wonder. It requires that a person pay close attention.”

In our writing prompt “ Do You See the Point in Learning Math? ” we wanted to know if students agreed. Basic arithmetic, sure, but is there value in learning higher-level math, such as algebra, geometry and calculus? Do we appreciate math enough?

The answer from many students — those who love and those who “detest” the subject alike — was yes. Of course math helps us balance checkbooks and work up budgets, they said, but it also helps us learn how to follow a formula, appreciate music, draw, shoot three-pointers and even skateboard. It gives us different perspectives, helps us organize our chaotic thoughts, makes us more creative, and shows us how to think rationally.

Not all were convinced that young people should have to take higher-level math classes all through high school, but, as one student said, “I can see myself understanding even more how important it is and appreciating it more as I get older.”

Thank you to all the teenagers who joined the conversation on our writing prompts this week, including students from Bentonville West High School in Centerton, Ark, ; Harvard-Westlake School in Los Angeles ; and North High School in North St. Paul, Minn.

Please note: Student comments have been lightly edited for length, but otherwise appear as they were originally submitted.

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What is Mathematics and Why Should Students Learn It?

When I am in the middle of a lesson, cooking along explaining things, working examples, perhaps rejoicing in the beauty of the subject matter (or my own perception of cleverness in thinking up a new analogy), there is always one question a student can ask that is guaranteed to throw me off my groove. That dreaded question is, “Why do we have to learn this?” As we get more seasoned and experienced as teachers, we perhaps learn ways to set things up in the beginning so that students are not tempted to ask this question. But even though I have been teaching for 16 years now, I still get it from time to time.

We may as well extend the question to all of mathematics. Why should students learn math at all beyond the simple skills needed to count change and pay bills? The need for learning more advanced mathematics may be obvious for students who will grow up to be scientists, engineers or financial officers. Naturally, one cannot do what an engineer has to do without a substantial background in advanced mathematics. But are algebra and geometry necessary for everybody? It is very easy to think, “Of course everyone has to take algebra! Everyone always takes algebra!” But our task is to see if this response can be justified.

For starters, we can probably all think of examples that illustrate just how challenging this question is. My own youngest daughter, now a senior in high school, struggles mightily with math and longs for the day when she can be done with it. She does feel bad about this, since I am her dad. Still, she wishes she lived in Jane Austen’s world, needing only to develop the “accomplishments” of a young lady, which happen to be the very things she loves— music, needlework, drawing, literature, and French. I, too, sometimes wish she could live in that world. That would be a nice life.

A completely different example is found in one of the great literary lights of the mid-twentieth century, Thomas Merton, author of The Seven Storey Mountain. Many of Merton’s formative years were spent in Europe, and as a youth Merton set his sights on studying at Cambridge. However, the very demanding exams he would have to take included mathematics that he had no talent for. He almost despaired of realizing his dream but then learned that he could avoid the math exams by even higher level achievement in the humanities, namely, studying his literature in the original languages and being examined accordingly. So, in addition to the classical languages he mastered and read Italian, French, and Spanish, passed the exams, and went to Cambridge.

It would be difficult indeed in contemporary times to design a school that can give prodigies like Merton what they need, and still be suitable for ordinary kids, as most of our students will be. I think I would be happy for any prodigy like Merton or Mozart to focus mainly on where his gift lies. I’m not going to worry about whether Mozart or Merton ever take algebra. But such prodigies are rare, and we must develop a rationale for our schools that applies to the other 99.999% of our students. The example of my daughter is probably a better example to challenge us as we address this question of why students should take mathematics. What about the ordinary kids? Why can’t girls be taught the way girls in Jane Austin novels were taught?

Before we develop a justification for including math in the curriculum, let’s pause for a moment to define the subject. To do this, I would like to make some observations about how the human mind works. Classical and Christian Education (CCE) schools typically emphasize the Trivium—grammar, logic, and rhetoric—and as a result students tend to exhibit above average performance in written and verbal expression through language. This is laudable, but interacting with the world and other people in the world through the written word represents only one part of human capability. The human mind is also wonderfully adept at imagining and discovering patterns, and communicating these symbolically. Moreover, as we have discovered during the past 400 years with the rise of contemporary science, our response to God’s creation is sometimes better facilitated by words, as in poetry or prose, and sometimes in the form of symbols, as in music and architecture. When the subject matter at hand deals with patterns, and with communicating ideas about specific patterns, communicating through the use of symbols is much more efficient than communicating through words.

This brings me to my definition of mathematics, a definition that is not original with me. I define mathematics as the study of patterns, a study that includes manipulating and expressing ideas about patterns symbolically and quantitatively. And though I will not be specifically addressing the Quadrivium in this essay, it seems to me that the key characteristic of the subjects in the Quadrivium, and the key thing to be preserved in education from the Quadrivium, is the centrality of searching for, identifying and describing patterns.

And now to our justification for including mathematics in the classical curriculum. Although it may sound strange to those espousing classical education, the first reason for teaching mathematics is the sheer practicality of well-developed mathematical skills. Please do not howl and stop up your ears; I am neither a modernist nor a utilitarian. But I ask, as I once was asked, “Should classical education be an ideal thing, or a realized thing?” Since we are here trying very hard to realize it at our schools, we must answer, of course, that it is to be a realized thing. Realizing any educational paradigm in any culture must involve the practical cultural question of who gets educated and why. In our culture it is not only the elite who get educated; it is everyone. This is a plain fact of democracy. We have no formal class system, we promote the freedom of the individual, and we have an educational system that has as its fundamental goal the broad education of the entire population so that every child has the opportunity to make his or her way in the world according to his or her own abilities and industry. In this country, in this century, education is for everyone and must serve the need for everyone to function in contemporary society. To do this, education must be practical. This means it will include living foreign languages, chemistry, and, of course, mathematics.

Practicality is defined by the age in which one lives. Practicality used to be about computing quantities of seed for planting, figuring sizes of parcels of land, or calculating exchange rates and unit quantities for commodities. Our high-tech age brings different requirements for the citizens. Nearly every job in the professional world, and many jobs in the trades as well, involve fairly sophisticated math. One doesn’t have to be an engineer to get into budget forecasting, statistical analysis of surveys, setting up spreadsheets, pre-tax paycheck deductions, network download rates, amortization, interest and tax calculations, cost vs. benefit analysis, the storage capacity of a back-up hard drive, and on and on.

Now, if practicality is one of the reasons for teaching math to everyone, it is also one of the criteria for determining what mathematics everyone should learn. When math is taught to everybody, practicality is a primary issue. This is why it is wise to require math studies to continue at least through Algebra 2 for all students possessing average or above average mathematical ability. Just as we expect everyone to gain a serviceable level of English proficiency for reading and writing, but do not expect everyone to be a writer, so in math we set the goal of a serviceable level of math proficiency suitable for life in the contemporary world, but do not expect everyone to be an expert in calculus. For many students this goal means that studies in math continue through Algebra 2, with perhaps some introductory statistics.

A student might reasonably argue that learning exponential decay functions or rules for powers and roots goes far beyond what is practical for most people. This is a reasonable point to make, and my response to it is two-fold. First, learning these more advanced skills in Algebra 2 is analogous to athletic training. Athletes train with arduous exercises, but this does not mean they will repeat these same actions in the game. The drills are demanding and are designed to get the athletes in shape so they can handle the actual game effectively. Similarly, we will expect that some mathematical topics and problems will be taught for their training value, and not because a particular type of relationship or function will be specifically needed in later life.

Second, contemporary issues constantly require citizens to think in quantitative terms, particularly in terms of a functional relationship between two or more variables. Mathematical relationships are now ubiquitous in modern society in every discussion of medicine, climate change, computer technologies, energy efficiencies, taxes, investments, survey results, profitability, trade, and so on. Exponential and power/root functions do come up all the time in particular fields of endeavor. But more generally, learning to handle them trains the mind to think in quantitative terms, with legitimate mathematical reasoning.

We are Americans living in America, and for 200 years Americans have been world-famous for their interest in practicality. If you want to get anyone’s attention in our culture today, including the professionals who are the parents of our students, you had better have a firm grip on the practical side of your discipline. Nowhere is this more true than in math and science. The competitive, high-tech, corporate-driven world we live in is unforgiving of weaknesses in math and science. If you can’t handle the math or the physics, there are plenty of students in developing countries who can, and they will take your place at the table and leave you to work your way up to an assistance manager’s job at Best Buy. Solid skill in math and science is very practical.

This brings me to one final point I wish to make on the practicality issue: without mastery (one of my favorite topics), no practical skill has been gained, and your efforts in the classroom have been in vain. Schools cranking out graduates that cannot do math are a dime a dozen. Our challenge in the CCE movement is to find a way to break through these decades of low performance into a new realm of proficiency and competence. Is this possible in a democracy? Ultimately, I do not know. But I think if we are wise in our efforts we will have our school families on our side as we do the hard work of developing a mastery-based curriculum.

So much for the practical value of teaching and learning mathematics. But while the modern world may be driven almost exclusively by the practical, for teachers in schools espousing a classical philosophy, the practical is not nearly enough. The reason for this is that as important as all the practical things are, they do not even come close to exhausting what being human is all about, and the core of the classical model of education is the goal of developing good human beings, not merely equipping people with practical trades.

Once we crack open the door on classical considerations for why we should study math we find that the reasons are just as extensive, if not more, as those on the practical side of the question. We could, for example, consider further my earlier point about the way the human mind works, and its capacity for expression in words as well as in mathematical symbols (as well as in the forms, colors and harmonies that are the raw materials of the arts). Or we could consider the matter from Plato’s point of view. In the Republic Plato taught that the proper subject for the education of a free man is that of being, the realm of the transcendent and permanent, as opposed to becoming, the realm of the temporal and transient. This was because he recognized in humans some kind of eternal, transcendent soul, and he viewed the proper task of education as feeding that transcendent soul. He saw mathematics as deeply connected to permanent, unchanging, transcendent truth, and thus a fitting subject for human beings to study. A third direction we could go would be to consider the Christian doctrine of the cultural mandate, and our understanding that Scripture charges God’s people with using Creation and all art, science and technology to improve the lives of fellow human beings, which is part of carrying out the Second Greatest Commandment. Finally, we could consider classical education from the point of view of pursuing truth, goodness and beauty as a means toward the development of wisdom and virtue.

For the present we will consider only the last of these possibilities, the pursuit of truth, goodness and beauty.

A good definition for classical education is the development of wisdom and virtue through the pursuit of truth, goodness and beauty. This ancient trilogy, reflected so vividly in Scripture in passages such as Philippians 4:8, focuses our attention on the deepest aspects of our humanity. G. K. Chesterton once wrote, “Art is the signature of man.” Creating or studying art requires the appreciation of truth, goodness, and beauty. Interestingly, so does making progress in fundamental scientific research. Let’s briefly consider truth, goodness and beauty and their relationship to mathematics each in turn.

The nature of truth has become clearer since the mid-twentieth century, for now we recognize that science and math are not so much concerned with discovering “truths” about the universe as they are modeling the universe. Students do not generally appreciate this until we lead them into discussion about it. Instead, they tend to take the findings of math and science as givens, as unchanging, eternal verities. But then we lead them to consider that science is not about discovering truth; it is about modeling the apparently infinite complexity of the natural world in an unending attempt to understand it better. And we lead them to understand that a similar principle applies to mathematics. The most glorious discoveries have been realized through learning the language of nature, mathematics, beautiful structures that can only be described mathematically, such as Maxwell’s Equations describing electromagnetism or Einstein’s General Theory of Relativity describing gravity.

But we also know that the connection between mathematics and truth is elusive. Kurt Gödel’s 1931 theorem demonstrated that mathematics can be consistent or comprehensive but not both. And before that the nineteenth century realization that Euclidean geometry was merely one convenient geometry among many geometries, and did not carry truth about the structure of the universe the way people had thought it did since the days of Euclid himself, brought many a philosopher to tears. If Euclidean geometry was not true, what was it? A great question; one we continue to explore. As I said, students do not appreciate these things unless we lead them into the discussion. However, once we distinguish these studies from truth itself and begin to use the arena of mathematics and science as a field for the continuing pursuit of truth, a deep and fruitful discussion begins.

Goodness is all around us in math and science for the simple reason that God declared his creation “good.” Thus, an element of our interaction with nature through math and science is the recognition that it is good that the apparent diameters of the sun and the moon as viewed from earth are nearly the same. It is good that the constant of proportionality in the relationship between mass and energy is simply the speed of light squared. It is good that the number of ancestors in each generation back from a given honeybee is given by the Fibonacci sequence. It is good that the planets’ orbits may be characterized accurately (though not exactly!) by Kepler’s Third Law of Planetary Motion. The double helix of our DNA with its multiple layers of instructional encoding and its capability for self-replication is inexpressibly good. So are the navigational abilities of migrating birds, the Doppler- shift detection capabilities of bat sonar, and the hexagonal shape of ice crystals. It is very good that all of nature displays a magnificent, sublime mathematical order that even non-Christian scientists have described as essentially miraculous. And it is very, very good that we humans have the cognitive ability to perceive and describe this order— these patterns—with mathematics. When students learn mathematics, the doors to see these things open before them. What could possibly be better than learning the language in which nature speaks to us, a language that enables us to behold the very goodness of God?

The third object of our pursuit as we develop wisdom and virtue is beauty. The relationship between mathematics and beauty is nothing short of mystical. It has been written about for ages, and illustrated in countless ways by countless writers, and yet we never tire of the subject. For many decades now scientists have recognized that the most valuable physical theories are those that are expressible in beautiful equations. Beauty has become a research tool, enabling us to probe the mathematical structure of the creation further and further. As with truth, leading students to see and appreciate the deep relationship between beauty and mathematics takes no small amount of effort. One has to begin by defining beauty. Then we have to establish the criteria we all use, usually subconsciously, when we make aesthetic judgments of all kinds. Finally, we have to demonstrate how these same aesthetic criteria apply in the domain of mathematics. As I said in the beginning, mathematics is the study of patterns, and patterns amaze and enthrall us with their beauty, from the patterns in a carbon nanostructure to those in the endlessly fascinating Mandelbrot Set. It is worth the effort to lead students to the point where they can consider and ponder beauty through the lens of mathematics.

Why should students study mathematics? We have seen that the study of mathematics is eminently practical, as practical as knowing how to read and write. And we have seen how mathematics provides a forum and a framework for the exploration of truth, goodness and beauty, a pursuit at the heart of our humanity and at the heart of the classical understanding of how humans should be educated. So on the question of students studying math, I think at this point it is safe to ask, why shouldn’t they?

Euclid’s Elements and Mathematics as a Liberal Art

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High School Mathematics at Work: Essays and Examples for the Education of All Students (1998)

Chapter: part one: connecting mathematics with work and life, part one— connecting mathematics with work and life.

Mathematics is the key to opportunity. No longer just the language of science, mathematics now contributes in direct and fundamental ways to business, finance, health, and defense. For students, it opens doors to careers. For citizens, it enables informed decisions. For nations, it provides knowledge to compete in a technological community. To participate fully in the world of the future, America must tap the power of mathematics. (NRC, 1989, p. 1)

The above statement remains true today, although it was written almost ten years ago in the Mathematical Sciences Education Board's (MSEB) report Everybody Counts (NRC, 1989). In envisioning a future in which all students will be afforded such opportunities, the MSEB acknowledges the crucial role played by formulae and algorithms, and suggests that algorithmic skills are more flexible, powerful, and enduring when they come from a place of meaning and understanding. This volume takes as a premise that all students can develop mathematical understanding by working with mathematical tasks from workplace and everyday contexts . The essays in this report provide some rationale for this premise and discuss some of the issues and questions that follow. The tasks in this report illuminate some of the possibilities provided by the workplace and everyday life.

Contexts from within mathematics also can be powerful sites for the development of mathematical understanding, as professional and amateur mathematicians will attest. There are many good sources of compelling problems from within mathematics, and a broad mathematics education will include experience with problems from contexts both within and outside mathematics. The inclusion of tasks in this volume is intended to highlight particularly compelling problems whose context lies outside of mathematics, not to suggest a curriculum.

The operative word in the above premise is "can." The understandings that students develop from any encounter with mathematics depend not only on the context, but also on the students' prior experience and skills, their ways of thinking, their engagement with the task, the environment in which they explore the task—including the teacher, the students, and the tools—the kinds of interactions that occur in that environment, and the system of internal and external incentives that might be associated with the activity. Teaching and learning are complex activities that depend upon evolving and rarely articulated interrelationships among teachers, students, materials, and ideas. No prescription for their improvement can be simple.

This volume may be beneficially seen as a rearticulation and elaboration of a principle put forward in Reshaping School Mathematics :

Principle 3: Relevant Applications Should be an Integral Part of the Curriculum.

Students need to experience mathematical ideas in the context in which they naturally arise—from simple counting and measurement to applications in business and science. Calculators and computers make it possible now to introduce realistic applications throughout the curriculum.

The significant criterion for the suitability of an application is whether it has the potential to engage students' interests and stimulate their mathematical thinking. (NRC, 1990, p. 38)

Mathematical problems can serve as a source of motivation for students if the problems engage students' interests and aspirations. Mathematical problems also can serve as sources of meaning and understanding if the problems stimulate students' thinking. Of course, a mathematical task that is meaningful to a student will provide more motivation than a task that does not make sense. The rationale behind the criterion above is that both meaning and motivation are required. The motivational benefits that can be provided by workplace and everyday problems are worth mentioning, for although some students are aware that certain mathematics courses are necessary in order to gain entry into particular career paths, many students are unaware of how particular topics or problem-solving approaches will have relevance in any workplace. The power of using workplace and everyday problems to teach mathematics lies not so much in motivation, however, for no con-

text by itself will motivate all students. The real power is in connecting to students' thinking.

There is growing evidence in the literature that problem-centered approaches—including mathematical contexts, "real world" contexts, or both—can promote learning of both skills and concepts. In one comparative study, for example, with a high school curriculum that included rich applied problem situations, students scored somewhat better than comparison students on algebraic procedures and significantly better on conceptual and problem-solving tasks (Schoen & Ziebarth, 1998). This finding was further verified through task-based interviews. Studies that show superior performance of students in problem-centered classrooms are not limited to high schools. Wood and Sellers (1996), for example, found similar results with second and third graders.

Research with adult learners seems to indicate that "variation of contexts (as well as the whole task approach) tends to encourage the development of general understanding in a way which concentrating on repeated routine applications of algorithms does not and cannot" (Strässer, Barr, Evans, & Wolf, 1991, p. 163). This conclusion is consistent with the notion that using a variety of contexts can increase the chance that students can show what they know. By increasing the number of potential links to the diverse knowledge and experience of the students, more students have opportunities to excel, which is to say that the above premise can promote equity in mathematics education.

There is also evidence that learning mathematics through applications can lead to exceptional achievement. For example, with a curriculum that emphasizes modeling and applications, high school students at the North Carolina School of Science and Mathematics have repeatedly submitted winning papers in the annual college competition, Mathematical Contest in Modeling (Cronin, 1988; Miller, 1995).

The relationships among teachers, students, curricular materials, and pedagogical approaches are complex. Nonetheless, the literature does supports the premise that workplace and everyday problems can enhance mathematical learning, and suggests that if students engage in mathematical thinking, they will be afforded opportunities for building connections, and therefore meaning and understanding.

In the opening essay, Dale Parnell argues that traditional teaching has been missing opportunities for connections: between subject-matter and context, between academic and vocational education, between school and life, between knowledge and application, and between subject-matter disciplines. He suggests that teaching must change if more students are to learn mathematics. The question, then, is how to exploit opportunities for connections between high school mathematics and the workplace and everyday life.

Rol Fessenden shows by example the importance of mathematics in business, specifically in making marketing decisions. His essay opens with a dialogue among employees of a company that intends to expand its business into

Japan, and then goes on to point out many of the uses of mathematics, data collection, analysis, and non-mathematical judgment that are required in making such business decisions.

In his essay, Thomas Bailey suggests that vocational and academic education both might benefit from integration, and cites several trends to support this suggestion: change and uncertainty in the workplace, an increased need for workers to understand the conceptual foundations of key academic subjects, and a trend in pedagogy toward collaborative, open-ended projects. Further-more, he observes that School-to-Work experiences, first intended for students who were not planning to attend a four-year college, are increasingly being seen as useful in preparing students for such colleges. He discusses several such programs that use work-related applications to teach academic skills and to prepare students for college. Integration of academic and vocational education, he argues, can serve the dual goals of "grounding academic standards in the realistic context of workplace requirements and introducing a broader view of the potential usefulness of academic skills even for entry level workers."

Noting the importance and utility of mathematics for jobs in science, health, and business, Jean Taylor argues for continued emphasis in high school of topics such as algebra, estimation, and trigonometry. She suggests that workplace and everyday problems can be useful ways of teaching these ideas for all students.

There are too many different kinds of workplaces to represent even most of them in the classrooms. Furthermore, solving mathematics problems from some workplace contexts requires more contextual knowledge than is reasonable when the goal is to learn mathematics. (Solving some other workplace problems requires more mathematical knowledge than is reasonable in high school.) Thus, contexts must be chosen carefully for their opportunities for sense making. But for students who have knowledge of a workplace, there are opportunities for mathematical connections as well. In their essay, Daniel Chazan and Sandra Callis Bethell describe an approach that creates such opportunities for students in an algebra course for 10th through 12th graders, many of whom carried with them a "heavy burden of negative experiences" about mathematics. Because the traditional Algebra I curriculum had been extremely unsuccessful with these students, Chazan and Bethell chose to do something different. One goal was to help students see mathematics in the world around them. With the help of community sponsors, Chazen and Bethell asked students to look for mathematics in the workplace and then describe that mathematics and its applications to their classmates.

The tasks in Part One complement the points made in the essays by making direct connections to the workplace and everyday life. Emergency Calls (p. 42) illustrates some possibilities for data analysis and representation by discussing the response times of two ambulance companies. Back-of-the-Envelope Estimates (p. 45) shows how quick, rough estimates and calculations

are useful for making business decisions. Scheduling Elevators (p. 49) shows how a few simplifying assumptions and some careful reasoning can be brought together to understand the difficult problem of optimally scheduling elevators in a large office building. Finally, in the context of a discussion with a client of an energy consulting firm, Heating-Degree-Days (p. 54) illuminates the mathematics behind a common model of energy consumption in home heating.

Cronin, T. P. (1988). High school students win "college" competition. Consortium: The Newsletter of the Consortium for Mathematics and Its Applications , 26 , 3, 12.

Miller, D. E. (1995). North Carolina sweeps MCM '94. SIAM News , 28 (2).

National Research Council. (1989). Everybody counts: A report to the nation on the future of mathematics education . Washington, DC: National Academy Press.

National Research Council. (1990). Reshaping school mathematics: A philosophy and framework for curriculum . Washington, DC: National Academy Press.

Schoen, H. L. & Ziebarth, S. W. (1998). Assessment of students' mathematical performance (A Core-Plus Mathematics Project Field Test Progress Report). Iowa City: Core Plus Mathematics Project Evaluation Site, University of Iowa.

Strässer, R., Barr, G. Evans, J. & Wolf, A. (1991). Skills versus understanding. In M. Harris (Ed.), Schools, mathematics, and work (pp. 158-168). London: The Falmer Press.

Wood, T. & Sellers, P. (1996). Assessment of a problem-centered mathematics program: Third grade. Journal for Research in Mathematics Education , 27 (3), 337-353.

1— Mathematics as a Gateway to Student Success

DALE PARNELL

Oregon State University

The study of mathematics stands, in many ways, as a gateway to student success in education. This is becoming particularly true as our society moves inexorably into the technological age. Therefore, it is vital that more students develop higher levels of competency in mathematics. 1

The standards and expectations for students must be high, but that is only half of the equation. The more important half is the development of teaching techniques and methods that will help all students (rather than just some students) reach those higher expectations and standards. This will require some changes in how mathematics is taught.

Effective education must give clear focus to connecting real life context with subject-matter content for the student, and this requires a more ''connected" mathematics program. In many of today's classrooms, especially in secondary school and college, teaching is a matter of putting students in classrooms marked "English," "history," or "mathematics," and then attempting to fill their heads with facts through lectures, textbooks, and the like. Aside from an occasional lab, workbook, or "story problem," the element of contextual teaching and learning is absent, and little attempt is made to connect what students are learning with the world in which they will be expected to work and spend their lives. Often the frag-

mented information offered to students is of little use or application except to pass a test.

What we do in most traditional classrooms is require students to commit bits of knowledge to memory in isolation from any practical application—to simply take our word that they "might need it later." For many students, "later" never arrives. This might well be called the freezer approach to teaching and learning. In effect, we are handing out information to our students and saying, "Just put this in your mental freezer; you can thaw it out later should you need it." With the exception of a minority of students who do well in mastering abstractions with little contextual experience, students aren't buying that offer. The neglected majority of students see little personal meaning in what they are asked to learn, and they just don't learn it.

I recently had occasion to interview 75 students representing seven different high schools in the Northwest. In nearly all cases, the students were juniors identified as vocational or general education students. The comment of one student stands out as representative of what most of these students told me in one way or another: "I know it's up to me to get an education, but a lot of times school is just so dull and boring. … You go to this class, go to that class, study a little of this and a little of that, and nothing connects. … I would like to really understand and know the application for what I am learning." Time and again, students were asking, "Why do I have to learn this?" with few sensible answers coming from the teachers.

My own long experience as a community college president confirms the thoughts of these students. In most community colleges today, one-third to one-half of the entering students are enrolled in developmental (remedial) education, trying to make up for what they did not learn in earlier education experiences. A large majority of these students come to the community college with limited mathematical skills and abilities that hardly go beyond adding, subtracting, and multiplying with whole numbers. In addition, the need for remediation is also experienced, in varying degrees, at four-year colleges and universities.

What is the greatest sin committed in the teaching of mathematics today? It is the failure to help students use the magnificent power of the brain to make connections between the following:

• subject-matter content and the context of use;
• academic and vocational education;
• school and other life experiences;
• knowledge and application of knowledge; and
• one subject-matter discipline and another.

Why is such failure so critical? Because understanding the idea of making the connection between subject-matter content and the context of application

is what students, at all levels of education, desperately require to survive and succeed in our high-speed, high-challenge, rapidly changing world.

Educational policy makers and leaders can issue reams of position papers on longer school days and years, site-based management, more achievement tests and better assessment practices, and other "hot" topics of the moment, but such papers alone will not make the crucial difference in what students know and can do. The difference will be made when classroom teachers begin to connect learning with real-life experiences in new, applied ways, and when education reformers begin to focus upon learning for meaning.

A student may memorize formulas for determining surface area and measuring angles and use those formulas correctly on a test, thereby achieving the behavioral objectives set by the teacher. But when confronted with the need to construct a building or repair a car, the same student may well be left at sea because he or she hasn't made the connection between the formulas and their real-life application. When students are asked to consider the Pythagorean Theorem, why not make the lesson active, where students actually lay out the foundation for a small building like a storage shed?

What a difference mathematics instruction could make for students if it were to stress the context of application—as well as the content of knowledge—using the problem-solving model over the freezer model. Teaching conducted upon the connected model would help more students learn with their thinking brain, as well as with their memory brain, developing the competencies and tools they need to survive and succeed in our complex, interconnected society.

One step toward this goal is to develop mathematical tasks that integrate subject-matter content with the context of application and that are aimed at preparing individuals for the world of work as well as for post-secondary education. Since many mathematics teachers have had limited workplace experience, they need many good examples of how knowledge of mathematics can be applied to real life situations. The trick in developing mathematical tasks for use in classrooms will be to keep the tasks connected to real life situations that the student will recognize. The tasks should not be just a contrived exercise but should stay as close to solving common problems as possible.

As an example, why not ask students to compute the cost of 12 years of schooling in a public school? It is a sad irony that after 12 years of schooling most students who attend the public schools have no idea of the cost of their schooling or how their education was financed. No wonder that some public schools have difficulty gaining financial support! The individuals being served by the schools have never been exposed to the real life context of who pays for the schools and why. Somewhere along the line in the teaching of mathematics, this real life learning opportunity has been missed, along with many other similar contextual examples.

The mathematical tasks in High School Mathematics at Work provide students (and teachers) with a plethora of real life mathematics problems and

challenges to be faced in everyday life and work. The challenge for teachers will be to develop these tasks so they relate as close as possible to where students live and work every day.

Parnell, D. (1985). The neglected majority . Washington, DC: Community College Press.

Parnell, D. (1995). Why do I have to learn this ? Waco, TX: CORD Communications.

D ALE P ARNELL is Professor Emeritus of the School of Education at Oregon State University. He has served as a University Professor, College President, and for ten years as the President and Chief Executive Officer of the American Association of Community Colleges. He has served as a consultant to the National Science Foundation and has served on many national commissions, such as the Secretary of Labor's Commission on Achieving Necessary Skills (SCANS). He is the author of the book The Neglected Majority which provided the foundation for the federally-funded Tech Prep Associate Degree Program.

2— Market Launch

ROL FESSENDEN

L. L. Bean, Inc.

"OK, the agenda of the meeting is to review the status of our launch into Japan. You can see the topics and presenters on the list in front of you. Gregg, can you kick it off with a strategy review?"

"Happy to, Bob. We have assessed the possibilities, costs, and return on investment of opening up both store and catalog businesses in other countries. Early research has shown that both Japan and Germany are good candidates. Specifically, data show high preference for good quality merchandise, and a higher-than-average propensity for an active outdoor lifestyle in both countries. Education, age, and income data are quite different from our target market in the U.S., but we do not believe that will be relevant because the cultures are so different. In addition, the Japanese data show that they have a high preference for things American, and, as you know, we are a classic American company. Name recognition for our company is 14%, far higher than any of our American competition in Japan. European competitors are virtually unrecognized, and other Far Eastern competitors are perceived to be of lower quality than us. The data on these issues are quite clear.

"Nevertheless, you must understand that there is a lot of judgment involved in the decision to focus on Japan. The analyses are limited because the cultures are different and we expect different behavioral drivers. Also,

much of the data we need in Japan are simply not available because the Japanese marketplace is less well developed than in the U.S. Drivers' license data, income data, lifestyle data, are all commonplace here and unavailable there. There is little prior penetration in either country by American retailers, so there is no experience we can draw upon. We have all heard how difficult it will be to open up sales operations in Japan, but recent sales trends among computer sellers and auto parts sales hint at an easing of the difficulties.

"The plan is to open three stores a year, 5,000 square feet each. We expect to do $700/square foot, which is more than double the experience of American retailers in the U.S. but 45% less than our stores. In addition, pricing will be 20% higher to offset the cost of land and buildings. Asset costs are approximately twice their rate in the U.S., but labor is slightly less. Benefits are more thoroughly covered by the government. Of course, there is a lot of uncertainty in the sales volumes we are planning. The pricing will cover some of the uncertainty but is still less than comparable quality goods already being offered in Japan.

"Let me shift over to the competition and tell you what we have learned. We have established long-term relationships with 500 to 1000 families in each country. This is comparable to our practice in the U.S. These families do not know they are working specifically with our company, as this would skew their reporting. They keep us appraised of their catalog and shopping experiences, regardless of the company they purchase from. The sample size is large enough to be significant, but, of course, you have to be careful about small differences.

"All the families receive our catalog and catalogs from several of our competitors. They match the lifestyle, income, and education demographic profiles of the people we want to have as customers. They are experienced catalog shoppers, and this will skew their feedback as compared to new catalog shoppers.

"One competitor is sending one 100-page catalog per quarter. The product line is quite narrow—200 products out of a domestic line of 3,000. They have selected items that are not likely to pose fit problems: primarily outerwear and knit shirts, not many pants, mostly men's goods, not women's. Their catalog copy is in Kanji, but the style is a bit stilted we are told, probably because it was written in English and translated, but we need to test this hypothesis. By contrast, we have simply mailed them the same catalog we use in the U.S., even written in English.

"Customer feedback has been quite clear. They prefer our broader assortment by a ratio of 3:1, even though they don't buy most of the products. As the competitors figured, sales are focused on outerwear and knits, but we are getting more sales, apparently because they like looking at the catalog and spend more time with it. Again, we need further testing. Another hypothesis is that our brand name is simply better known.

"Interestingly, they prefer our English-language version because they find it more of an adventure to read the catalog in another language. This is probably

a built-in bias of our sampling technique because we specifically selected people who speak English. We do not expect this trend to hold in a general mailing.

"The English language causes an 8% error rate in orders, but orders are 25% larger, and 4% more frequent. If we can get them to order by phone, we can correct the errors immediately during the call.

"The broader assortment, as I mentioned, is resulting in a significantly higher propensity to order, more units per order, and the same average unit cost. Of course, paper and postage costs increase as a consequence of the larger format catalog. On the other hand, there are production efficiencies from using the same version as the domestic catalog. Net impact, even factoring in the error rate, is a significant sales increase. On the other hand, most of the time, the errors cause us to ship the wrong item which then needs to be mailed back at our expense, creating an impression in the customers that we are not well organized even though the original error was theirs.

"Final point: The larger catalog is being kept by the customer an average of 70 days, while the smaller format is only kept on average for 40 days. Assuming—we need to test this—that the length of time they keep the catalog is proportional to sales volumes, this is good news. We need to assess the overall impact carefully, but it appears that there is a significant population for which an English-language version would be very profitable."

"Thanks, Gregg, good update. Jennifer, what do you have on customer research?"

"Bob, there's far more that we need to know than we have been able to find out. We have learned that Japan is very fad-driven in apparel tastes and fascinated by American goods. We expect sales initially to sky-rocket, then drop like a stone. Later on, demand will level out at a profitable level. The graphs on page 3 [ Figure 2-1 ] show demand by week for 104 weeks, and we have assessed several scenarios. They all show a good underlying business, but the uncertainty is in the initial take-off. The best data are based on the Italian fashion boom which Japan experienced in the late 80s. It is not strictly analogous because it revolved around dress apparel instead of our casual and weekend wear. It is, however, the best information available.

FIGURE 2-1: Sales projections by week, Scenario A

FIGURE 2-2: Size distributions, U.S. vs. Japan

"Our effectiveness in positioning inventory for that initial surge will be critical to our long-term success. There are excellent data—supplied by MITI, I might add—that show that Japanese customers can be intensely loyal to companies that meet their high service expectations. That is why we prepared several scenarios. Of course, if we position inventory for the high scenario, and we experience the low one, we will experience a significant loss due to liquidations. We are still analyzing the long-term impact, however. It may still be worthwhile to take the risk if the 2-year ROI 1 is sufficient.

"We have solid information on their size scales [ Figure 2-2 ]. Seventy percent are small and medium. By comparison, 70% of Americans are large and extra large. This will be a challenge to manage but will save a few bucks on fabric.

"We also know their color preferences, and they are very different than Americans. Our domestic customers are very diverse in their tastes, but 80% of Japanese customers will buy one or two colors out of an offering of 15. We are still researching color choices, but it varies greatly for pants versus shirts, and for men versus women. We are confident we can find patterns, but we also know that it is easy to guess wrong in that market. If we guess wrong, the liquidation costs will be very high.

"Bad news on the order-taking front, however. They don't like to order by phone. …"

In this very brief exchange among decision-makers we observe the use of many critically important skills that were originally learned in public schools. Perhaps the most important is one not often mentioned, and that is the ability to convert an important business question into an appropriate mathematical one, to solve the mathematical problem, and then to explain the implications of the solution for the original business problem. This ability to inhabit simultaneously the business world and the mathematical world, to translate between the two, and, as a consequence, to bring clarity to complex, real-world issues is of extraordinary importance.

In addition, the participants in this conversation understood and interpreted graphs and tables, computed, approximated, estimated, interpolated, extrapolated, used probabilistic concepts to draw conclusions, generalized from

small samples to large populations, identified the limits of their analyses, discovered relationships, recognized and used variables and functions, analyzed and compared data sets, and created and interpreted models. Another very important aspect of their work was that they identified additional questions, and they suggested ways to shed light on those questions through additional analysis.

There were two broad issues in this conversation that required mathematical perspectives. The first was to develop as rigorous and cost effective a data collection and analysis process as was practical. It involved perhaps 10 different analysts who attacked the problem from different viewpoints. The process also required integration of the mathematical learnings of all 10 analysts and translation of the results into business language that could be understood by non-mathematicians.

The second broad issue was to understand from the perspective of the decision-makers who were listening to the presentation which results were most reliable, which were subject to reinterpretation, which were actually judgments not supported by appropriate analysis, and which were hypotheses that truly required more research. In addition, these business people would likely identify synergies in the research that were not contemplated by the analysts. These synergies need to be analyzed to determine if—mathematically—they were real. The most obvious one was where the inventory analysts said that the customers don't like to use the phone to place orders. This is bad news for the sales analysts who are counting on phone data collection to correct errors caused by language problems. Of course, we need more information to know the magnitude—or even the existance—of the problem.

In brief, the analyses that preceded the dialogue might each be considered a mathematical task in the business world:

• A cost analysis of store operations and catalogs was conducted using data from existing American and possibly other operations.
• Customer preferences research was analyzed to determine preferences in quality and life-style. The data collection itself could not be carried out by a high school graduate without guidance, but 80% of the analysis could.
• Cultural differences were recognized as a causes of analytical error. Careful analysis required judgment. In addition, sources of data were identified in the U.S., and comparable sources were found lacking in Japan. A search was conducted for other comparable retail experience, but none was found. On the other hand, sales data from car parts and computers were assessed for relevance.
• Rates of change are important in understanding how Japanese and American stores differ. Sales per square foot, price increases,
• asset costs, labor costs and so forth were compared to American standards to determine whether a store based in Japan would be a viable business.
• "Nielsen" style ratings of 1000 families were used to collect data. Sample size and error estimates were mentioned. Key drivers of behavior (lifestyle, income, education) were mentioned, but this list may not be complete. What needs to be known about these families to predict their buying behavior? What does "lifestyle" include? How would we quantify some of these variables?
• A hypothesis was presented that catalog size and product diversity drive higher sales. What do we need to know to assess the validity of this hypothesis? Another hypothesis was presented about the quality of the translation. What was the evidence for this hypothesis? Is this a mathematical question? Sales may also be proportional to the amount of time a potential customer retains the catalog. How could one ascertain this?
• Despite the abundance of data, much uncertainty remains about what to expect from sales over the first two years. Analysis could be conducted with the data about the possible inventory consequences of choosing the wrong scenario.
• One might wonder about the uncertainty in size scales. What is so difficult about identifying the colors that Japanese people prefer? Can these preferences be predicted? Will this increase the complexity of the inventory management task?
• Can we predict how many people will not use phones? What do they use instead?

As seen through a mathematical lens, the business world can be a rich, complex, and essentially limitless source of fascinating questions.

R OL F ESSENDEN is Vice-President of Inventory Planning and Control at L. L. Bean, Inc. He is also Co-Principal Investigator and Vice-Chair of Maine's State Systemic Initiative and Chair of the Strategic Planning Committee. He has previously served on the Mathematical Science Education Board, and on the National Alliance for State Science and Mathematics Coalitions (NASSMC).

3— Integrating Vocational and Academic Education

THOMAS BAILEY

Columbia University

In high school education, preparation for work immediately after high school and preparation for post-secondary education have traditionally been viewed as incompatible. Work-bound high-school students end up in vocational education tracks, where courses usually emphasize specific skills with little attention to underlying theoretical and conceptual foundations. 1 College-bound students proceed through traditional academic discipline-based courses, where they learn English, history, science, mathematics, and foreign languages, with only weak and often contrived references to applications of these skills in the workplace or in the community outside the school. To be sure, many vocational teachers do teach underlying concepts, and many academic teachers motivate their lessons with examples and references to the world outside the classroom. But these enrichments are mostly frills, not central to either the content or pedagogy of secondary school education.

Rethinking Vocational and Academic Education

Educational thinking in the United States has traditionally placed priority on college preparation. Thus the distinct track of vocational education has been seen as an option for those students who are deemed not capable of success in the more desirable academic track. As vocational programs acquired a reputation

as a ''dumping ground," a strong background in vocational courses (especially if they reduced credits in the core academic courses) has been viewed as a threat to the college aspirations of secondary school students.

This notion was further reinforced by the very influential 1983 report entitled A Nation at Risk (National Commission on Excellence in Education, 1983), which excoriated the U.S. educational system for moving away from an emphasis on core academic subjects that, according to the report, had been the basis of a previously successful American education system. Vocational courses were seen as diverting high school students from core academic activities. Despite the dubious empirical foundation of the report's conclusions, subsequent reforms in most states increased the number of academic courses required for graduation and reduced opportunities for students to take vocational courses.

The distinction between vocational students and college-bound students has always had a conceptual flaw. The large majority of students who go to four-year colleges are motivated, at least to a significant extent, by vocational objectives. In 1994, almost 247,000 bachelors degrees were conferred in business administration. That was only 30,000 less than the total number (277,500) of 1994 bachelor degree conferred in English, mathematics, philosophy, religion, physical sciences and science technologies, biological and life sciences, social sciences, and history combined . Furthermore, these "academic" fields are also vocational since many students who graduate with these degrees intend to make their living working in those fields.

Several recent economic, technological, and educational trends challenge this sharp distinction between preparation for college and for immediate post-high-school work, or, more specifically, challenge the notion that students planning to work after high school have little need for academic skills while college-bound students are best served by an abstract education with only tenuous contact with the world of work:

• First, many employers and analysts are arguing that, due to changes in the nature of work, traditional approaches to teaching vocational skills may not be effective in the future. Given the increasing pace of change and uncertainty in the workplace, young people will be better prepared, even for entry level positions and certainly for subsequent positions, if they have an underlying understanding of the scientific, mathematical, social, and even cultural aspects of the work that they will do. This has led to a growing emphasis on integrating academic and vocational education. 2
• Views about teaching and pedagogy have increasingly moved toward a more open and collaborative "student-centered" or "constructivist" teaching style that puts a great deal of emphasis on having students work together on complex, open-ended projects. This reform strategy is now widely implemented through the efforts of organizations such as the Coalition of Essential Schools, the National Center for Restructuring Education, Schools, and Teaching at
• Teachers College, and the Center for Education Research at the University of Wisconsin at Madison. Advocates of this approach have not had much interaction with vocational educators and have certainly not advocated any emphasis on directly preparing high school students for work. Nevertheless, the approach fits well with a reformed education that integrates vocational and academic skills through authentic applications. Such applications offer opportunities to explore and combine mathematical, scientific, historical, literary, sociological, economic, and cultural issues.
• In a related trend, the federal School-to-Work Opportunities Act of 1994 defines an educational strategy that combines constructivist pedagogical reforms with guided experiences in the workplace or other non-work settings. At its best, school-to-work could further integrate academic and vocational learning through appropriately designed experiences at work.
• The integration of vocational and academic education and the initiatives funded by the School-to-Work Opportunities Act were originally seen as strategies for preparing students for work after high school or community college. Some educators and policy makers are becoming convinced that these approaches can also be effective for teaching academic skills and preparing students for four-year college. Teaching academic skills in the context of realistic and complex applications from the workplace and community can provide motivational benefits and may impart a deeper understanding of the material by showing students how the academic skills are actually used. Retention may also be enhanced by giving students a chance to apply the knowledge that they often learn only in the abstract. 3
• During the last twenty years, the real wages of high school graduates have fallen and the gap between the wages earned by high school and college graduates has grown significantly. Adults with no education beyond high school have very little chance of earning enough money to support a family with a moderate lifestyle. 4 Given these wage trends, it seems appropriate and just that every high school student at least be prepared for college, even if some choose to work immediately after high school.

Innovative Examples

There are many examples of programs that use work-related applications both to teach academic skills and to prepare students for college. One approach is to organize high school programs around broad industrial or occupational areas, such as health, agriculture, hospitality, manufacturing, transportation, or the arts. These broad areas offer many opportunities for wide-ranging curricula in all academic disciplines. They also offer opportunities for collaborative work among teachers from different disciplines. Specific skills can still be taught in this format but in such a way as to motivate broader academic and theoretical themes. Innovative programs can now be found in many vocational

high schools in large cities, such as Aviation High School in New York City and the High School of Agricultural Science and Technology in Chicago. Other schools have organized schools-within-schools based on broad industry areas.

Agriculturally based activities, such as 4H and Future Farmers of America, have for many years used the farm setting and students' interest in farming to teach a variety of skills. It takes only a little imagination to think of how to use the social, economic, and scientific bases of agriculture to motivate and illustrate skills and knowledge from all of the academic disciplines. Many schools are now using internships and projects based on local business activities as teaching tools. One example among many is the integrated program offered by the Thomas Jefferson High School for Science and Technology in Virginia, linking biology, English, and technology through an environmental issues forum. Students work as partners with resource managers at the Mason Neck National Wildlife Refuge and the Mason Neck State Park to collect data and monitor the daily activities of various species that inhabit the region. They search current literature to establish a hypothesis related to a real world problem, design an experiment to test their hypothesis, run the experiment, collect and analyze data, draw conclusions, and produce a written document that communicates the results of the experiment. The students are even responsible for determining what information and resources are needed and how to access them. Student projects have included making plans for public education programs dealing with environmental matters, finding solutions to problems caused by encroaching land development, and making suggestions for how to handle the overabundance of deer in the region.

These examples suggest the potential that a more integrated education could have for all students. Thus continuing to maintain a sharp distinction between vocational and academic instruction in high school does not serve the interests of many of those students headed for four-year or two-year college or of those who expect to work after high school. Work-bound students will be better prepared for work if they have stronger academic skills, and a high-quality curriculum that integrates school-based learning into work and community applications is an effective way to teach academic skills for many students.

Despite the many examples of innovative initiatives that suggest the potential for an integrated view, the legacy of the duality between vocational and academic education and the low status of work-related studies in high school continue to influence education and education reform. In general, programs that deviate from traditional college-prep organization and format are still viewed with suspicion by parents and teachers focused on four-year college. Indeed, college admissions practices still very much favor the traditional approaches. Interdisciplinary courses, "applied" courses, internships, and other types of work experience that characterize the school-to-work strategy or programs that integrate academic and vocational education often do not fit well into college admissions requirements.

Joining Work and Learning

What implications does this have for the mathematics standards developed by the National Council of Teachers of Mathematics (NCTM)? The general principle should be to try to design standards that challenge rather than reinforce the distinction between vocational and academic instruction. Academic teachers of mathematics and those working to set academic standards need to continue to try to understand the use of mathematics in the workplace and in everyday life. Such understandings would offer insights that could suggest reform of the traditional curriculum, but they would also provide a better foundation for teaching mathematics using realistic applications. The examples in this volume are particularly instructive because they suggest the importance of problem solving, logic, and imagination and show that these are all important parts of mathematical applications in realistic work settings. But these are only a beginning.

In order to develop this approach, it would be helpful if the NCTM standards writers worked closely with groups that are setting industry standards. 5 This would allow both groups to develop a deeper understanding of the mathematics content of work.

The NCTM's Curriculum Standards for Grades 9-12 include both core standards for all students and additional standards for "college-intending" students. The argument presented in this essay suggests that the NCTM should dispense with the distinction between college intending and non-college intending students. Most of the additional standards, those intended only for the "college intending" students, provide background that is necessary or beneficial for the calculus sequence. A re-evaluation of the role of calculus in the high school curriculum may be appropriate, but calculus should not serve as a wedge to separate college-bound from non-college-bound students. Clearly, some high school students will take calculus, although many college-bound students will not take calculus either in high school or in college. Thus in practice, calculus is not a characteristic that distinguishes between those who are or are not headed for college. Perhaps standards for a variety of options beyond the core might be offered. Mathematics standards should be set to encourage stronger skills for all students and to illustrate the power and usefulness of mathematics in many settings. They should not be used to institutionalize dubious distinctions between groups of students.

Bailey, T. & Merritt, D. (1997). School-to-work for the collegebound . Berkeley, CA: National Center for Research in Vocational Education.

Hoachlander, G . (1997) . Organizing mathematics education around work . In L.A. Steen (Ed.), Why numbers count: Quantitative literacy for tomorrow's America , (pp. 122-136). New York: College Entrance Examination Board.

Levy, F. & Murnane, R. (1992). U.S. earnings levels and earnings inequality: A review of recent trends and proposed explanations. Journal of Economic Literature , 30 , 1333-1381.

National Commission on Excellence in Education. (1983). A nation at risk: The imperative for educational reform . Washington, DC: Author.

T HOMAS B AILEY is an Associate Professor of Economics Education at Teachers College, Columbia University. He is also Director of the Institute on Education and the Economy and Director of the Community College Research Center, both at Teachers College. He is also on the board of the National Center for Research in Vocational Education.

4— The Importance of Workplace and Everyday Mathematics

JEAN E. TAYLOR

Rutgers University

For decades our industrial society has been based on fossil fuels. In today's knowledge-based society, mathematics is the energy that drives the system. In the words of the new WQED television series, Life by the Numbers , to create knowledge we "burn mathematics." Mathematics is more than a fixed tool applied in known ways. New mathematical techniques and analyses and even conceptual frameworks are continually required in economics, in finance, in materials science, in physics, in biology, in medicine.

Just as all scientific and health-service careers are mathematically based, so are many others. Interaction with computers has become a part of more and more jobs, and good analytical skills enhance computer use and troubleshooting. In addition, virtually all levels of management and many support positions in business and industry require some mathematical understanding, including an ability to read graphs and interpret other information presented visually, to use estimation effectively, and to apply mathematical reasoning.

What Should Students Learn for Today's World?

Education in mathematics and the ability to communicate its predictions is more important than ever for moving from low-paying jobs into better-paying ones. For example, my local paper, The Times of Trenton , had a section "Focus

on Careers" on October 5, 1997 in which the majority of the ads were for high technology careers (many more than for sales and marketing, for example).

But precisely what mathematics should students learn in school? Mathematicians and mathematics educators have been discussing this question for decades. This essay presents some thoughts about three areas of mathematics—estimation, trigonometry, and algebra—and then some thoughts about teaching and learning.

Estimation is one of the harder skills for students to learn, even if they experience relatively little difficulty with other aspects of mathematics. Many students think of mathematics as a set of precise rules yielding exact answers and are uncomfortable with the idea of imprecise answers, especially when the degree of precision in the estimate depends on the context and is not itself given by a rule. Yet it is very important to be able to get an approximate sense of the size an answer should be, as a way to get a rough check on the accuracy of a calculation (I've personally used it in stores to detect that I've been charged twice for the same item, as well as often in my own mathematical work), a feasibility estimate, or as an estimation for tips.

Trigonometry plays a significant role in the sciences and can help us understand phenomena in everyday life. Often introduced as a study of triangle measurement, trigonometry may be used for surveying and for determining heights of trees, but its utility extends vastly beyond these triangular applications. Students can experience the power of mathematics by using sine and cosine to model periodic phenomena such as going around and around a circle, going in and out with tides, monitoring temperature or smog components changing on a 24-hour cycle, or the cycling of predator-prey populations.

No educator argues the importance of algebra for students aiming for mathematically-based careers because of the foundation it provides for the more specialized education they will need later. Yet, algebra is also important for those students who do not currently aspire to mathematics-based careers, in part because a lack of algebraic skills puts an upper bound on the types of careers to which a student can aspire. Former civil rights leader Robert Moses makes a good case for every student learning algebra, as a means of empowering students and providing goals, skills, and opportunities. The same idea was applied to learning calculus in the movie Stand and Deliver . How, then, can we help all students learn algebra?

For me personally, the impetus to learn algebra was at least in part to learn methods of solution for puzzles. Suppose you have 39 jars on three shelves. There are twice as many jars on the second shelf as the first, and four more jars on the third shelf than on the second shelf. How many jars are there on each shelf? Such problems are not important by themselves, but if they show the students the power of an idea by enabling them to solve puzzles that they'd like to solve, then they have value. We can't expect such problems to interest all students. How then can we reach more students?

Workplace and Everyday Settings as a Way of Making Sense

One of the common tools in business and industry for investigating mathematical issues is the spreadsheet, which is closely related to algebra. Writing a rule to combine the elements of certain cells to produce the quantity that goes into another cell is doing algebra, although the variables names are cell names rather than x or y . Therefore, setting up spreadsheet analyses requires some of the thinking that algebra requires.

By exploring mathematics via tasks which come from workplace and everyday settings, and with the aid of common tools like spreadsheets, students are more likely to see the relevance of the mathematics and are more likely to learn it in ways that are personally meaningful than when it is presented abstractly and applied later only if time permits. Thus, this essay argues that workplace and everyday tasks should be used for teaching mathematics and, in particular, for teaching algebra. It would be a mistake, however, to rely exclusively on such tasks, just as it would be a mistake to teach only spreadsheets in place of algebra.

Communicating the results of an analysis is a fundamental part of any use of mathematics on a job. There is a growing emphasis in the workplace on group work and on the skills of communicating ideas to colleagues and clients. But communicating mathematical ideas is also a powerful tool for learning, for it requires the student to sharpen often fuzzy ideas.

Some of the tasks in this volume can provide the kinds of opportunities I am talking about. Another problem, with clear connections to the real world, is the following, taken from the book entitled Consider a Spherical Cow: A Course in Environmental Problem Solving , by John Harte (1988). The question posed is: How does biomagnification of a trace substance occur? For example, how do pesticides accumulate in the food chain, becoming concentrated in predators such as condors? Specifically, identify the critical ecological and chemical parameters determining bioconcentrations in a food chain, and in terms of these parameters, derive a formula for the concentration of a trace substance in each link of a food chain. This task can be undertaken at several different levels. The analysis in Harte's book is at a fairly high level, although it still involves only algebra as a mathematical tool. The task could be undertaken at a more simple level or, on the other hand, it could be elaborated upon as suggested by further exercises given in that book. And the students could then present the results of their analyses to each other as well as the teacher, in oral or written form.

Concepts or Procedures?

When teaching mathematics, it is easy to spend so much time and energy focusing on the procedures that the concepts receive little if any attention. When teaching algebra, students often learn the procedures for using the quadratic formula or for solving simultaneous equations without thinking of intersections of curves and lines and without being able to apply the procedures in unfamiliar settings. Even

when concentrating on word problems, students often learn the procedures for solving "coin problems" and "train problems" but don't see the larger algebraic context. The formulas and procedures are important, but are not enough.

When using workplace and everyday tasks for teaching mathematics, we must avoid falling into the same trap of focusing on the procedures at the expense of the concepts. Avoiding the trap is not easy, however, because just like many tasks in school algebra, mathematically based workplace tasks often have standard procedures that can be used without an understanding of the underlying mathematics. To change a procedure to accommodate a changing business climate, to respond to changes in the tax laws, or to apply or modify a procedure to accommodate a similar situation, however, requires an understanding of the mathematical ideas behind the procedures. In particular, a student should be able to modify the procedures for assessing energy usage for heating (as in Heating-Degree-Days, p. 54) in order to assess energy usage for cooling in the summer.

To prepare our students to make such modifications on their own, it is important to focus on the concepts as well as the procedures. Workplace and everyday tasks can provide opportunities for students to attach meaning to the mathematical calculations and procedures. If a student initially solves a problem without algebra, then the thinking that went into his or her solution can help him or her make sense out of algebraic approaches that are later presented by the teacher or by other students. Such an approach is especially appropriate for teaching algebra, because our teaching of algebra needs to reach more students (too often it is seen by students as meaningless symbol manipulation) and because algebraic thinking is increasingly important in the workplace.

An Example: The Student/Professor Problem

To illustrate the complexity of learning algebra meaningfully, consider the following problem from a study by Clement, Lockhead, & Monk (1981):

Write an equation for the following statement: "There are six times as many students as professors at this university." Use S for the number of students and P for the number of professors. (p. 288)

The authors found that of 47 nonscience majors taking college algebra, 57% got it wrong. What is more surprising, however, is that of 150 calculus-level students, 37% missed the problem.

A first reaction to the most common wrong answer, 6 S = P , is that the students simply translated the words of the problems into mathematical symbols without thinking more deeply about the situation or the variables. (The authors note that some textbooks instruct students to use such translation.)

By analyzing transcripts of interviews with students, the authors found this approach and another (faulty) approach, as well. These students often drew a diagram showing six students and one professor. (Note that we often instruct students to draw diagrams when solving word problems.) Reasoning

from the diagram, and regarding S and P as units, the student may write 6 S = P , just as we would correctly write 12 in. = 1 ft. Such reasoning is quite sensible, though it misses the fundamental intent in the problem statement that S is to represent the number of students, not a student.

Thus, two common suggestions for students—word-for-word translation and drawing a diagram—can lead to an incorrect answer to this apparently simple problem, if the students do not more deeply contemplate what the variables are intended to represent. The authors found that students who wrote and could explain the correct answer, S = 6 P , drew upon a richer understanding of what the equation and the variables represent.

Clearly, then, we must encourage students to contemplate the meanings of variables. Yet, part of the power and efficiency of algebra is precisely that one can manipulate symbols independently of what they mean and then draw meaning out of the conclusions to which the symbolic manipulations lead. Thus, stable, long-term learning of algebraic thinking requires both mastery of procedures and also deeper analytical thinking.

Paradoxically, the need for sharper analytical thinking occurs alongside a decreased need for routine arithmetic calculation. Calculators and computers make routine calculation easier to do quickly and accurately; cash registers used in fast food restaurants sometimes return change; checkout counters have bar code readers and payment takes place by credit cards or money-access cards.

So it is education in mathematical thinking, in applying mathematical computation, in assessing whether an answer is reasonable, and in communicating the results that is essential. Teaching mathematics via workplace and everyday problems is an approach that can make mathematics more meaningful for all students. It is important, however, to go beyond the specific details of a task in order to teach mathematical ideas. While this approach is particularly crucial for those students intending to pursue careers in the mathematical sciences, it will also lead to deeper mathematical understanding for all students.

Clement, J., Lockhead, J., & Monk, G. (1981). Translation difficulties in learning mathematics. American Mathematical Monthly , 88 , 286-290.

Harte, J. (1988). Consider a spherical cow: A course in environmental problem solving . York, PA: University Science Books.

J EAN E. T AYLOR is Professor of Mathematics at Rutgers, the State University of New Jersey. She is currently a member of the Board of Directors of the American Association for the Advancement of Science and formerly chaired its Section A Nominating Committee. She has served as Vice President and as a Member-at-Large of the Council of the American Mathematical Society, and served on its Executive Committee and its Nominating Committee. She has also been a member of the Joint Policy Board for Mathematics, and a member of the Board of Advisors to The Geometry Forum (now The Mathematics Forum) and to the WQED television series, Life by the Numbers .

5— Working with Algebra

DANIEL CHAZAN

Michigan State University

SANDRA CALLIS BETHELL

Holt High School

Teaching a mathematics class in which few of the students have demonstrated success is a difficult assignment. Many teachers avoid such assignments, when possible. On the one hand, high school mathematics teachers, like Bertrand Russell, might love mathematics and believe something like the following:

Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. … Remote from human passions, remote even from the pitiful facts of nature, the generations have gradually created an ordered cosmos, where pure thought can dwell as in its nature home, and where one, at least, of our nobler impulses can escape from the dreary exile of the natural world. (Russell, 1910, p. 73)

But, on the other hand, students may not have the luxury, in their circumstances, of appreciating this beauty. Many of them may not see themselves as thinkers because contemplation would take them away from their primary

focus: how to get by in a world that was not created for them. Instead, like Jamaica Kincaid, they may be asking:

What makes the world turn against me and all who look like me? I won nothing, I survey nothing, when I ask this question, the luxury of an answer that will fill volumes does not stretch out before me. When I ask this question, my voice is filled with despair. (Kincaid, 1996, pp. 131-132)

Our Teaching and Issues it Raised

During the 1991-92 and 1992-93 school years, we (a high school teacher and a university teacher educator) team taught a lower track Algebra I class for 10th through 12th grade students. 1 Most of our students had failed mathematics before, and many needed to pass Algebra I in order to complete their high school mathematics requirement for graduation. For our students, mathematics had become a charged subject; it carried a heavy burden of negative experiences. Many of our students were convinced that neither they nor their peers could be successful in mathematics.

Few of our students did well in other academic subjects, and few were headed on to two- or four-year colleges. But the students differed in their affiliation with the high school. Some, called ''preppies" or "jocks" by others, were active participants in the school's activities. Others, "smokers" or "stoners," were rebelling to differing degrees against school and more broadly against society. There were strong tensions between members of these groups. 2

Teaching in this setting gives added importance and urgency to the typical questions of curriculum and motivation common to most algebra classes. In our teaching, we explored questions such as the following:

• What is it that we really want high school students, especially those who are not college-intending, to study in algebra and why?
• What is the role of algebra's manipulative skills in a world with graphing calculators and computers? How do the manipulative skills taught in the traditional curriculum give students a new perspective on, and insight into, our world?
• If our teaching efforts depend on students' investment in learning, on what grounds can we appeal to them, implicitly or explicitly, for energy and effort? In a tracked, compulsory setting, how can we help students, with broad interests and talents and many of whom are not college-intending, see value in a shared exploration of algebra?

An Approach to School Algebra

As a result of thinking about these questions, in our teaching we wanted to avoid being in the position of exhorting students to appreciate the beauty or utility of algebra. Our students were frankly skeptical of arguments based on

utility. They saw few people in their community using algebra. We had also lost faith in the power of extrinsic rewards and punishments, like failing grades. Many of our students were skeptical of the power of the high school diploma to alter fundamentally their life circumstances. We wanted students to find the mathematical objects we were discussing in the world around them and thus learn to value the perspective that this mathematics might give them on their world.

To help us in this task, we found it useful to take what we call a "relationships between quantities" approach to school algebra. In this approach, the fundamental mathematical objects of study in school algebra are functions that can be represented by inputs and outputs listed in tables or sketched or plotted on graphs, as well as calculation procedures that can be written with algebraic symbols. 3 Stimulated, in part, by the following quote from August Comte, we viewed these functions as mathematical representations of theories people have developed for explaining relationships between quantities.

In the light of previous experience, we must acknowledge the impossibility of determining, by direct measurement, most of the heights and distances we should like to know. It is this general fact which makes the science of mathematics necessary. For in renouncing the hope, in almost every case, of measuring great heights or distances directly, the human mind has had to attempt to determine them indirectly, and it is thus that philosophers were led to invent mathematics. (Quoted in Serres, 1982, p. 85)

The "Sponsor" Project

Using this approach to the concept of function, during the 1992-93 school year, we designed a year-long project for our students. The project asked pairs of students to find the mathematical objects we were studying in the workplace of a community sponsor. Students visited the sponsor's workplace four times during the year—three after-school visits and one day-long excused absence from school. In these visits, the students came to know the workplace and learned about the sponsor's work. We then asked students to write a report describing the sponsor's workplace and answering questions about the nature of the mathematical activity embedded in the workplace. The questions are organized in Table 5-1 .

Using These Questions

In order to determine how the interviews could be structured and to provide students with a model, we chose to interview Sandra's husband, John Bethell, who is a coatings inspector for an engineering firm. When asked about his job, John responded, "I argue for a living." He went on to describe his daily work inspecting contractors painting water towers. Since most municipalities contract with the lowest bidder when a water tower needs to be painted, they will often hire an engineering firm to make sure that the contractor works according to specification. Since the contractor has made a low bid, there are strong

TABLE 5-1: Questions to ask in the workplace

financial incentives for the contractor to compromise on quality in order to make a profit.

In his work John does different kinds of inspections. For example, he has a magnetic instrument to check the thickness of the paint once it has been applied to the tower. When it gives a "thin" reading, contractors often question the technology. To argue for the reading, John uses the surface area of the tank, the number of paint cans used, the volume of paint in the can, and an understanding of the percentage of this volume that evaporates to calculate the average thickness of the dry coating. Other examples from his workplace involve the use of tables and measuring instruments of different kinds.

Some Examples of Students' Work

When school started, students began working on their projects. Although many of the sponsors initially indicated that there were no mathematical dimensions to their work, students often were able to show sponsors places where the mathematics we were studying was to be found. For example, Jackie worked with a crop and soil scientist. She was intrigued by the way in which measurement of weight is used to count seeds. First, her sponsor would weigh a test batch of 100 seeds to generate a benchmark weight. Then, instead of counting a large number of seeds, the scientist would weigh an amount of seeds and compute the number of seeds such a weight would contain.

Rebecca worked with a carpeting contractor who, in estimating costs, read the dimensions of rectangular rooms off an architect's blueprint, multiplied to find the area of the room in square feet (doing conversions where necessary), then multiplied by a cost per square foot (which depended on the type of carpet) to compute the cost of the carpet. The purpose of these estimates was to prepare a bid for the architect where the bid had to be as low as possible without making the job unprofitable. Rebecca used a chart ( Table 5-2 ) to explain this procedure to the class.

Joe and Mick, also working in construction, found out that in laying pipes, there is a "one by one" rule of thumb. When digging a trench for the placement of the pipe, the non-parallel sides of the trapezoidal cross section must have a slope of 1 foot down for every one foot across. This ratio guarantees that the dirt in the hole will not slide down on itself. Thus, if at the bottom of the hole, the trapezoid must have a certain width in order to fit the pipe, then on ground level the hole must be this width plus twice the depth of the hole. Knowing in advance how wide the hole must be avoids lengthy and costly trial and error.

Other students found that functions were often embedded in cultural artifacts found in the workplace. For example, a student who visited a doctor's office brought in an instrument for predicting the due dates of pregnant women, as well as providing information about average fetal weight and length ( Figure 5-1 ).

TABLE 5-2: Cost of carpet worksheet

FIGURE 5-1: Pregnancy wheel

While the complexities of organizing this sort of project should not be minimized—arranging sponsors, securing parental permission, and meeting administrators and parent concerns about the requirement of off-campus, after-school work—we remain intrigued by the potential of such projects for helping students see mathematics in the world around them. The notions of identifying central mathematical objects for a course and then developing ways of identifying those objects in students' experience seems like an important alternative to the use of application-based materials written by developers whose lives and social worlds may be quite different from those of students.

Chazen, D. (1996). Algebra for all students? Journal of Mathematical Behavior , 15 (4), 455-477.

Eckert, P. (1989). Jocks and burnouts: Social categories and identity in the high school . New York: Teachers College Press.

Fey, J. T., Heid, M. K., et al. (1995). Concepts in algebra: A technological approach . Dedham, MA: Janson Publications.

Kieran, C., Boileau, A., & Garancon, M. (1996). Introducing algebra by mean of a technology-supported, functional approach. In N. Bednarz et al. (Eds.), Approaches to algebra , (pp. 257-293). Kluwer Academic Publishers: Dordrecht, The Netherlands.

Kincaid, J. (1996). The autobiography of my mother . New York: Farrar, Straus, Giroux.

Nemirovsky, R. (1996). Mathematical narratives, modeling and algebra. In N. Bednarz et al. (Eds.) Approaches to algebra , (pp. 197-220). Kluwer Academic Publishers: Dordrecht, The Netherlands.

Russell, B. (1910). Philosophical Essays . London: Longmans, Green.

Schwartz, J. & Yerushalmy, M. (1992). Getting students to function in and with algebra. In G. Harel & E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy , (MAA Notes, Vol. 25, pp. 261-289). Washington, DC: Mathematical Association of America.

Serres, M. (1982). Mathematics and philosophy: What Thales saw … In J. Harari & D. Bell (Eds.), Hermes: Literature, science, philosophy , (pp. 84-97). Baltimore, MD: Johns Hopkins.

Thompson, P. (1993). Quantitative reasoning, complexity, and additive structures. Educational Studies in Mathematics , 25 , 165-208.

Yerushalmy, M. & Schwartz, J. L. (1993). Seizing the opportunity to make algebra mathematically and pedagogically interesting. In T. A. Romberg, E. Fennema, & T. P. Carpenter (Eds.), Integrating research on the graphical representation of functions , (pp. 41-68). Hillsdale, NJ: Lawrence Erlbaum Associates.

D ANIEL C HAZAN is an Associate Professor of Teacher Education at Michigan State University. To assist his research in mathematics teaching and learning, he has taught algebra at the high school level. His interests include teaching mathematics by examining student ideas, using computers to support student exploration, and the potential for the history and philosophy of mathematics to inform teaching.

S ANDRA C ALLIS B ETHELL has taught mathematics and Spanish at Holt High School for 10 years. She has also completed graduate work at Michigan State University and Western Michigan University. She has interest in mathematics reform, particularly in meeting the needs of diverse learners in algebra courses.

Emergency Calls

A city is served by two different ambulance companies. City logs record the date, the time of the call, the ambulance company, and the response time for each 911 call ( Table 1 ). Analyze these data and write a report to the City Council (with supporting charts and graphs) advising it on which ambulance company the 911 operators should choose to dispatch for calls from this region.

TABLE 1: Ambulance dispatch log sheet, May 1–30

This problem confronts the student with a realistic situation and a body of data regarding two ambulance companies' response times to emergency calls. The data the student is provided are typically "messy"—just a log of calls and response times, ordered chronologically. The question is how to make sense of them. Finding patterns in data such as these requires a productive mixture of mathematics common sense, and intellectual detective work. It's the kind of reasoning that students should be able to do—the kind of reasoning that will pay off in the real world.

Mathematical Analysis

In this case, a numerical analysis is not especially informative. On average, the companies are about the same: Arrow has a mean response time of 11.4 minutes compared to 11.6 minutes for Metro. The spread of the data is also not very helpful. The ranges of their distributions are exactly the same: from 6 minutes to 19 minutes. The standard deviation of Arrow's response time is a little longer—4.3 minutes versus 3.4 minutes for Metro—indicating that Arrow's response times fluctuate a bit more.

Graphs of the response times (Figures 1 and 2 ) reveal interesting features. Both companies, especially Arrow, seem to have bimodal distributions, which is to say that there are two clusters of data without much data in between.

FIGURE 1: Distribution of Arrow's response times

FIGURE 2: Distribution of Metro's response times

The distributions for both companies suggest that there are some other factors at work. Might a particular driver be the problem? Might the slow response times for either company be on particular days of the week or at particular times of day? Graphs of the response time versus the time of day (Figures 3 and 4 ) shed some light on these questions.

FIGURE 3: Arrow response times by time of day

FIGURE 4: Metro response times by time of day

These graphs show that Arrow's response times were fast except between 5:30 AM and 9:00 AM, when they were about 9 minutes slower on average. Similarly, Metro's response times were fast except between about 3:30 PM and 6:30 PM, when they were about 5 minutes slower. Perhaps the locations of the companies make Arrow more susceptible to the morning rush hour and Metro more susceptible to the afternoon rush hour. On the other hand, the employees on Arrow's morning shift or Metro's afternoon shift may not be efficient. To avoid slow responses, one could recommend to the City Council that Metro be called during the morning and that Arrow be called during the afternoon. A little detective work into the sources of the differences between the companies may yield a better recommendation.

Comparisons may be drawn between two samples in various contexts—response times for various services (taxis, computer-help desks, 24-hour hot lines at automobile manufacturers) being one class among many. Depending upon the circumstances, the data may tell very different stories. Even in the situation above, if the second pair of graphs hadn't offered such clear explanations, one might have argued that although the response times for Arrow were better on average the spread was larger, thus making their "extremes" more risky. The fundamental idea is using various analysis and representation techniques to make sense of data when the important factors are not necessarily known ahead of time.

Back-of-the-Envelope Estimates

Practice "back-of-the-envelope" estimates based on rough approximations that can be derived from common sense or everyday observations. Examples:

• Consider a public high school mathematics teacher who feels that students should work five nights a week, averaging about 35 minutes a night, doing focused on-task work and who intends to grade all homework with comments and corrections. What is a reasonable number of hours per week that such a teacher should allocate for grading homework?
• How much paper does The New York Times use in a week? A paper company that wishes to make a bid to become their sole supplier needs to know whether they have enough current capacity. If the company were to store a two-week supply of newspaper, will their empty 14,000 square foot warehouse be big enough?

Some 50 years ago, physicist Enrico Fermi asked his students at the University of Chicago, "How many piano tuners are there in Chicago?" By asking such questions, Fermi wanted his students to make estimates that involved rough approximations so that their goal would be not precision but the order of magnitude of their result. Thus, many people today call these kinds of questions "Fermi questions." These generally rough calculations often require little more than common sense, everyday observations, and a scrap of paper, such as the back of a used envelope.

Scientists and mathematicians use the idea of order of magnitude , usually expressed as the closest power of ten, to give a rough sense of the size of a quantity. In everyday conversation, people use a similar idea when they talk about "being in the right ballpark." For example, a full-time job at minimum wage yields an annual income on the order of magnitude of $10,000 or 10 4 dollars. Some corporate executives and professional athletes make annual salaries on the order of magnitude of $10,000,000 or 10 7 dollars. To say that these salaries differ by a factor of 1000 or 10 3 , one can say that they differ by three orders of magnitude. Such a lack of precision might seem unscientific or unmathematical, but such approximations are quite useful in determining whether a more precise measurement is feasible or necessary, what sort of action might be required, or whether the result of a calculation is "in the right ballpark." In choosing a strategy to protect an endangered species, for example, scientists plan differently if there are 500 animals remaining than if there are 5,000. On the other hand, determining whether 5,200 or 6,300 is a better estimate is not necessary, as the strategies will probably be the same.

Careful reasoning with everyday observations can usually produce Fermi estimates that are within an order of magnitude of the exact answer (if there is one). Fermi estimates encourage students to reason creatively with approximate quantities and uncertain information. Experiences with such a process can help an individual function in daily life to determine the reasonableness of numerical calculations, of situations or ideas in the workplace, or of a proposed tax cut. A quick estimate of some revenue- or profit-enhancing scheme may show that the idea is comparable to suggesting that General Motors enter the summer sidewalk lemonade market in your neighborhood. A quick estimate could encourage further investigation or provide the rationale to dismiss the idea.

Almost any numerical claim may be treated as a Fermi question when the problem solver does not have access to all necessary background information. In such a situation, one may make rough guesses about relevant numbers, do a few calculations, and then produce estimates.

The examples are solved separately below.

Grading Homework

Although many component factors vary greatly from teacher to teacher or even from week to week, rough calculations are not hard to make. Some important factors to consider for the teacher are: how many classes he or she teaches, how many students are in each of the classes, how much experience has the teacher had in general and has the teacher previously taught the classes, and certainly, as part of teaching style, the kind of homework the teacher assigns, not to mention the teacher's efficiency in grading.

Suppose the teacher has 5 classes averaging 25 students per class. Because the teacher plans to write corrections and comments, assume that the students' papers contain more than a list of answers—they show some student work and, perhaps, explain some of the solutions. Grading such papers might take as long as 10 minutes each, or perhaps even longer. Assuming that the teacher can grade them as quickly as 3 minutes each, on average, the teacher's grading time is:

This is an impressively large number, especially for a teacher who already spends almost 25 hours/week in class, some additional time in preparation, and some time meeting with individual students. Is it reasonable to expect teachers to put in that kind of time? What compromises or other changes might the teacher make to reduce the amount of time? The calculation above offers four possibilities: Reduce the time spent on each homework paper, reduce the number of students per class, reduce the number of classes taught each day, or reduce the number of days per week that homework will be collected. If the teacher decides to spend at most 2 hours grading each night, what is the total number of students for which the teacher should have responsibility? This calculation is a partial reverse of the one above:

If the teacher still has 5 classes, that would mean 8 students per class!

The New York Times

Answering this question requires two preliminary estimates: the circulation of The New York Times and the size of the newspaper. The answers will probably be different on Sundays. Though The New York Times is a national newspaper, the number of subscribers outside the New York metropolitan area is probably small compared to the number inside. The population of the New York metropolitan area is roughly ten million people. Since most families buy at most one copy, and not all families buy The New York Times , the circulation might be about 1 million newspapers each day. (A circulation of 500,000 seems too small and 2 million seems too big.) The Sunday and weekday editions probably have different

circulations, but assume that they are the same since they probably differ by less than a factor of two—much less than an order of magnitude. When folded, a weekday edition of the paper measures about 1/2 inch thick, a little more than 1 foot long, and about 1 foot wide. A Sunday edition of the paper is the same width and length, but perhaps 2 inches thick. For a week, then, the papers would stack 6 × 1/2 + 2 = 5 inches thick, for a total volume of about 1 ft × 1 ft × 5/12 ft = 0.5 ft 3 .

The whole circulation, then, would require about 1/2 million cubic feet of paper per week, or about 1 million cubic feet for a two-week supply.

Is the company's warehouse big enough? The paper will come on rolls, but to make the estimates easy, assume it is stacked. If it were stacked 10 feet high, the supply would require 100,000 square feet of floor space. The company's 14,000 square foot storage facility will probably not be big enough as its size differs by almost an order of magnitude from the estimate. The circulation estimate and the size of the newspaper estimate should each be within a factor of 2, implying that the 100,000 square foot estimate is off by at most a factor of 4—less than an order of magnitude.

How big a warehouse is needed? An acre is 43,560 square feet so about two acres of land is needed. Alternatively, a warehouse measuring 300 ft × 300 ft (the length of a football field in both directions) would contain 90,000 square feet of floor space, giving a rough idea of the size.

After gaining some experience with these types of problems, students can be encouraged to pay close attention to the units and to be ready to make and support claims about the accuracy of their estimates. Paying attention to units and including units as algebraic quantities in calculations is a common technique in engineering and the sciences. Reasoning about a formula by paying attention only to the units is called dimensional analysis.

Sometimes, rather than a single estimate, it is helpful to make estimates of upper and lower bounds. Such an approach reinforces the idea that an exact answer is not the goal. In many situations, students could first estimate upper and lower bounds, and then collect some real data to determine whether the answer lies between those bounds. In the traditional game of guessing the number of jelly beans in a jar, for example, all students should be able to estimate within an order of magnitude, or perhaps within a factor of two. Making the closest guess, however, involves some chance.

Fermi questions are useful outside the workplace. Some Fermi questions have political ramifications:

• How many miles of streets are in your city or town? The police chief is considering increasing police presence so that every street is patrolled by car at least once every 4 hours.
• When will your town fill up its landfill? Is this a very urgent matter for the town's waste management personnel to assess in depth?
• In his 1997 State of the Union address, President Clinton renewed his call for a tax deduction of up to $10,000 for the cost of college tuition. He estimates that 16.5 million students stand to benefit. Is this a reasonable estimate of the number who might take advantage of the tax deduction? How much will the deduction cost in lost federal revenue?

Creating Fermi problems is easy. Simply ask quantitative questions for which there is no practical way to determine exact values. Students could be encouraged to make up their own. Examples are: ''How many oak trees are there in Illinois?" or "How many people in the U.S. ate chicken for dinner last night?" "If all the people in the world were to jump in the ocean, how much would it raise the water level?" Give students the opportunity to develop their own Fermi problems and to share them with each other. It can stimulate some real mathematical thinking.

Scheduling Elevators

In some buildings, all of the elevators can travel to all of the floors, while in others the elevators are restricted to stopping only on certain floors. What is the advantage of having elevators that travel only to certain floors? When is this worth instituting?

Scheduling elevators is a common example of an optimization problem that has applications in all aspects of business and industry. Optimal scheduling in general not only can save time and money, but it can contribute to safety (e.g., in the airline industry). The elevator problem further illustrates an important feature of many economic and political arguments—the dilemma of trying simultaneously to optimize several different needs.

Politicians often promise policies that will be the least expensive, save the most lives, and be best for the environment. Think of flood control or occupational safety rules, for example. When we are lucky, we can perhaps find a strategy of least cost, a strategy that saves the most lives, or a strategy that damages the environment least. But these might not be the same strategies: generally one cannot simultaneously satisfy two or more independent optimization conditions. This is an important message for students to learn, in order to become better educated and more critical consumers and citizens.

In the elevator problem, customer satisfaction can be emphasized by minimizing the average elevator time (waiting plus riding) for employees in an office building. Minimizing wait-time during rush hours means delivering many people quickly, which might be accomplished by filling the elevators and making few stops. During off-peak hours, however, minimizing wait-time means maximizing the availability of the elevators. There is no reason to believe that these two goals will yield the same strategy. Finding the best strategy for each is a mathematical problem; choosing one of the two strategies or a compromise strategy is a management decision, not a mathematical deduction.

This example serves to introduce a complex topic whose analysis is well within the range of high school students. Though the calculations require little more than arithmetic, the task puts a premium on the creation of reasonable alternative strategies. Students should recognize that some configurations (e.g., all but one elevator going to the top floor and the one going to all the others) do not merit consideration, while others are plausible. A systematic evaluation of all possible configurations is usually required to find the optimal solution. Such a systematic search of the possible solution space is important in many modeling situations where a formal optimal strategy is not known. Creating and evaluating reasonable strategies for the elevators is quite appropriate for high school student mathematics and lends itself well to thoughtful group effort. How do you invent new strategies? How do you know that you have considered all plausible strategies? These are mathematical questions, and they are especially amenable to group discussion.

Students should be able to use the techniques first developed in solving a simple case with only a few stories and a few elevators to address more realistic situations (e.g., 50 stories, five elevators). Using the results of a similar but simpler problem to model a more complicated problem is an important way to reason in mathematics. Students

need to determine what data and variables are relevant. Start by establishing the kind of building—a hotel, an office building, an apartment building? How many people are on the different floors? What are their normal destinations (e.g., primarily the ground floor or, perhaps, a roof-top restaurant). What happens during rush hours?

To be successful at the elevator task, students must first develop a mathematical model of the problem. The model might be a graphical representation for each elevator, with time on the horizontal axis and the floors represented on the vertical axis, or a tabular representation indicating the time spent on each floor. Students must identify the pertinent variables and make simplifying assumptions about which of the possible floors an elevator will visit.

This section works through some of the details in a particularly simple case. Consider an office building with six occupied floors, employing 240 people, and a ground floor that is not used for business. Suppose there are three elevators, each of which can hold 10 people. Further suppose that each elevator takes approximately 25 seconds to fill on the ground floor, then takes 5 seconds to move between floors and 15 seconds to open and close at each floor on which it stops.

Scenario One

What happens in the morning when everyone arrives for work? Assume that everyone arrives at approximately the same time and enters the elevators on the ground floor. If all elevators go to all floors and if the 240 people are evenly divided among all three elevators, each elevator will have to make 8 trips of 10 people each.

When considering a single trip of one elevator, assume for simplicity that 10 people get on the elevator at the ground floor and that it stops at each floor on the way up, because there may be an occupant heading to each floor. Adding 5 seconds to move to each floor and 15 seconds to stop yields 20 seconds for each of the six floors. On the way down, since no one is being picked up or let off, the elevator does not stop, taking 5 seconds for each of six floors for a total of 30 seconds. This round-trip is represented in Table 1 .

TABLE 1: Elevator round-trip time, Scenario one

Since each elevator makes 8 trips, the total time will be 1,400 seconds or 23 minutes, 20 seconds.

Scenario Two

Now suppose that one elevator serves floors 1–3 and, because of the longer trip, two elevators are assigned to floors 4–6. The elevators serving the top

TABLE 2: Elevator round-trip times, Scenario two

floors will save 15 seconds for each of floors 1–3 by not stopping. The elevator serving the bottom floors will save 20 seconds for each of the top floors and will save time on the return trip as well. The times for these trips are shown in Table 2 .

Assuming the employees are evenly distributed among the floors (40 people per floor), elevator A will transport 120 people, requiring 12 trips, and elevators B and C will transport 120 people, requiring 6 trips each. These trips will take 1200 seconds (20 minutes) for elevator A and 780 seconds (13 minutes) for elevators B and C, resulting in a small time savings (about 3 minutes) over the first scenario. Because elevators B and C are finished so much sooner than elevator A, there is likely a more efficient solution.

Scenario Three

The two round-trip times in Table 2 do not differ by much because the elevators move quickly between floors but stop at floors relatively slowly. This observation suggests that a more efficient arrangement might be to assign each elevator to a pair of floors. The times for such a scenario are listed in Table 3 .

Again assuming 40 employees per floor, each elevator will deliver 80 people, requiring 8 trips, taking at most a total of 920 seconds. Thus this assignment of elevators results in a time savings of almost 35% when compared with the 1400 seconds it would take to deliver all employees via unassigned elevators.

TABLE 3: Elevator round-trip times, Scenario three

Perhaps this is the optimal solution. If so, then the above analysis of this simple case suggests two hypotheses:

• The optimal solution assigns each floor to a single elevator.
• If the time for stopping is sufficiently larger than the time for moving between floors, each elevator should serve the same number of floors.

Mathematically, one could try to show that this solution is optimal by trying all possible elevator assignments or by carefully reasoning, perhaps by showing that the above hypotheses are correct. Practically, however, it doesn't matter because this solution considers only the morning rush hour and ignores periods of low use.

The assignment is clearly not optimal during periods of low use, and much of the inefficiency is related to the first hypothesis for rush hour optimization: that each floor is served by a single elevator. With this condition, if an employee on floor 6 arrives at the ground floor just after elevator C has departed, for example, she or he will have to wait nearly two minutes for elevator C to return, even if elevators A and B are idle. There are other inefficiencies that are not considered by focusing on the rush hour. Because each floor is served by a single elevator, an employee who wishes to travel from floor 3 to floor 6, for example, must go via the ground floor and switch elevators. Most employees would prefer more flexibility than a single elevator serving each floor.

At times when the elevators are not all busy, unassigned elevators will provide the quickest response and the greatest flexibility.

Because this optimal solution conflicts with the optimal rush hour solution, some compromise is necessary. In this simple case, perhaps elevator A could serve all floors, elevator B could serve floors 1-3, and elevator C could serve floors 4-6.

The second hypothesis, above, deserves some further thought. The efficiency of the rush hour solution Table 3 is due in part to the even division of employees among the floors. If employees were unevenly distributed with, say, 120 of the 240 people working on the top two floors, then elevator C would need to make 12 trips, taking a total of 1380 seconds, resulting in almost no benefit over unassigned elevators. Thus, an efficient solution in an actual building must take into account the distribution of the employees among the floors.

Because the stopping time on each floor is three times as large as the traveling time between floors (15 seconds versus 5 seconds), this solution effectively ignores the traveling time by assigning the same number of employees to each elevator. For taller buildings, the traveling time will become more significant. In those cases fewer employees should be assigned to the elevators that serve the upper floors than are assigned to the elevators that serve the lower floors.

The problem can be made more challenging by altering the number of elevators, the number of floors, and the number of individuals working on each floor. The rate of movement of elevators can be determined by observing buildings in the local area. Some elevators move more quickly than others. Entrance and exit times could also be measured by students collecting

data on local elevators. In a similar manner, the number of workers, elevators, and floors could be taken from local contexts.

A related question is, where should the elevators go when not in use? Is it best for them to return to the ground floor? Should they remain where they were last sent? Should they distribute themselves evenly among the floors? Or should they go to floors of anticipated heavy traffic? The answers will depend on the nature of the building and the time of day. Without analysis, it will not be at all clear which strategy is best under specific conditions. In some buildings, the elevators are controlled by computer programs that "learn" and then anticipate the traffic patterns in the building.

A different example that students can easily explore in detail is the problem of situating a fire station or an emergency room in a city. Here the key issue concerns travel times to the region being served, with conflicting optimization goals: average time vs. maximum time. A location that minimizes the maximum time of response may not produce the least average time of response. Commuters often face similar choices in selecting routes to work. They may want to minimize the average time, the maximum time, or perhaps the variance, so that their departure and arrival times are more predictable.

Most of the optimization conditions discussed so far have been expressed in units of time. Sometimes, however, two optimization conditions yield strategies whose outcomes are expressed in different (and sometimes incompatible) units of measurement. In many public policy issues (e.g., health insurance) the units are lives and money. For environmental issues, sometimes the units themselves are difficult to identify (e.g., quality of life).

When one of the units is money, it is easy to find expensive strategies but impossible to find ones that have virtually no cost. In some situations, such as airline safety, which balances lives versus dollars, there is no strategy that minimize lives lost (since additional dollars always produce slight increases in safety), and the strategy that minimizes dollars will be at $0. Clearly some compromise is necessary. Working with models of different solutions can help students understand the consequences of some of the compromises.

Heating-Degree-Days

An energy consulting firm that recommends and installs insulation and similar energy saving devices has received a complaint from a customer. Last summer she paid $540 to insulate her attic on the prediction that it would save 10% on her natural gas bills. Her gas bills have been higher than the previous winter, however, and now she wants a refund on the cost of the insulation. She admits that this winter has been colder than the last, but she had expected still to see some savings.

The facts: This winter the customer has used 1,102 therms, whereas last winter she used only 1,054 therms. This winter has been colder: 5,101 heating-degree-days this winter compared to 4,201 heating-degree-days last winter. (See explanation below.) How does a representative of the energy consulting firm explain to this customer that the accumulated heating-degree-days measure how much colder this winter has been, and then explain how to calculate her anticipated versus her actual savings.

Explaining the mathematics behind a situation can be challenging and requires a real knowledge of the context, the procedures, and the underlying mathematical concepts. Such communication of mathematical ideas is a powerful learning device for students of mathematics as well as an important skill for the workplace. Though the procedure for this problem involves only proportions, a thorough explanation of the mathematics behind the procedure requires understanding of linear modeling and related algebraic reasoning, accumulation and other precursors of calculus, as well as an understanding of energy usage in home heating.

The customer seems to understand that a straight comparison of gas usage does not take into account the added costs of colder weather, which can be significant. But before calculating any anticipated or actual savings, the customer needs some understanding of heating-degree-days. For many years, weather services and oil and gas companies have been using heating-degree-days to explain and predict energy usage and to measure energy savings of insulation and other devices. Similar degree-day units are also used in studying insect populations and crop growth. The concept provides a simple measure of the accumulated amount of cold or warm weather over time. In the discussion that follows, all temperatures are given in degrees Fahrenheit, although the process is equally workable using degrees Celsius.

Suppose, for example, that the minimum temperature in a city on a given day is 52 degrees and the maximum temperature is 64 degrees. The average temperature for the day is then taken to be 58 degrees. Subtracting that result from 65 degrees (the cutoff point for heating), yields 7 heating-degree-days for the day. By recording high and low temperatures and computing their average each day, heating-degree-days can be accumulated over the course of a month, a winter, or any period of time as a measure of the coldness of that period.

Over five consecutive days, for example, if the average temperatures were 58, 50, 60, 67, and 56 degrees Fahrenheit, the calculation yields 7, 15, 5, 0, and 9 heating-degree-days respectively, for a total accumulation of 36 heating-degree-days for the five days. Note that the fourth day contributes 0 heating-degree-days to the total because the temperature was above 65 degrees.

The relationship between average temperatures and heating-degree-days is represented graphically in Figure 1 . The average temperatures are shown along the solid line graph. The area of each shaded rectangle represents the number of heating-degree-days for that day, because the width of each rectangle is one day and the height of each rectangle is the number of degrees below 65 degrees. Over time, the sum of the areas of the rectangles represents the number of heating-degree-days accumulated during the period. (Teachers of calculus will recognize connections between these ideas and integral calculus.)

The statement that accumulated heating-degree-days should be proportional to gas or heating oil usage is based primarily on two assumptions: first, on a day for which the average temperature is above 65 degrees, no heating should be required, and therefore there should be no gas or heating oil usage; second, a day for which the average temperature is 25 degrees (40 heating-degree-days) should require twice as much heating as a day for which the average temperature is 45

FIGURE 1: Daily heating-degree-days

degrees (20 heating-degree-days) because there is twice the temperature difference from the 65 degree cutoff.

The first assumption is reasonable because most people would not turn on their heat if the temperature outside is above 65 degrees. The second assumption is consistent with Newton's law of cooling, which states that the rate at which an object cools is proportional to the difference in temperature between the object and its environment. That is, a house which is 40 degrees warmer than its environment will cool at twice the rate (and therefore consume energy at twice the rate to keep warm) of a house which is 20 degrees warmer than its environment.

The customer who accepts the heating-degree-day model as a measure of energy usage can compare this winter's usage with that of last winter. Because 5,101/4,201 = 1.21, this winter has been 21% colder than last winter, and therefore each house should require 21% more heat than last winter. If this customer hadn't installed the insulation, she would have required 21% more heat than last year, or about 1,275 therms. Instead, she has required only 5% more heat (1,102/1,054 = 1.05), yielding a savings of 14% off what would have been required (1,102/1,275 = .86).

Another approach to this would be to note that last year the customer used 1,054 therms/4,201 heating-degree-days = .251 therms/heating-degree-day, whereas this year she has used 1,102 therms/5,101 heating-degree-days = .216 therms/heating-degree-day, a savings of 14%, as before.

How good is the heating-degree-day model in predicting energy usage? In a home that has a thermometer and a gas meter or a gauge on a tank, students could record daily data for gas usage and high and low temperature to test the accuracy of the model. Data collection would require only a few minutes per day for students using an electronic indoor/outdoor thermometer that tracks high and low temperatures. Of course, gas used for cooking and heating water needs to be taken into account. For homes in which the gas tank has no gauge or doesn't provide accurate enough data, a similar experiment could be performed relating accumulated heating-degree-days to gas or oil usage between fill-ups.

It turns out that in well-sealed modern houses, the cutoff temperature for heating can be lower than 65 degrees (sometimes as low as 55 degrees) because of heat generated by light bulbs, appliances, cooking, people, and pets. At temperatures sufficiently below the cutoff, linearity turns out to be a good assumption. Linear regression on the daily usage data (collected as suggested above) ought to find an equation something like U = -.251( T - 65), where T is the average temperature and U is the gas usage. Note that the slope, -.251, is the gas usage per heating-degree-day, and 65 is the cutoff. Note also that the accumulation of heating-degree-days takes a linear equation and turns it into a proportion. There are some important data analysis issues that could be addressed by such an investigation. It is sometimes dangerous, for example, to assume linearity with only a few data points, yet this widely used model essentially assumes linearity from only one data point, the other point having coordinates of 65 degrees, 0 gas usage.

Over what range of temperatures, if any, is this a reasonable assumption? Is the standard method of computing average temperature a good method? If, for example, a day is mostly near 20 degrees but warms up to 50 degrees for a short time in the afternoon, is 35 heating-degree-days a good measure of the heating required that day? Computing averages of functions over time is a standard problem that can be solved with integral calculus. With knowledge of typical and extreme rates of temperature change, this could become a calculus problem or a problem for approximate solution by graphical methods without calculus, providing background experience for some of the important ideas in calculus.

Students could also investigate actual savings after insulating a home in their school district. A customer might typically see 8-10% savings for insulating roofs, although if the house is framed so that the walls act like chimneys, ducting air from the house and the basement into the attic, there might be very little savings. Eliminating significant leaks, on the other hand, can yield savings of as much as 25%.

Some U.S. Department of Energy studies discuss the relationship between heating-degree-days and performance and find the cutoff temperature to be lower in some modern houses. State energy offices also have useful documents.

What is the relationship between heating-degree-days computed using degrees Fahrenheit, as above, and heating-degree-days computed using degrees Celsius? Showing that the proper conversion is a direct proportion and not the standard Fahrenheit-Celsius conversion formula requires some careful and sophisticated mathematical thinking.

Traditionally, vocational mathematics and precollege mathematics have been separate in schools. But the technological world in which today's students will work and live calls for increasing connection between mathematics and its applications. Workplace-based mathematics may be good mathematics for everyone.

High School Mathematics at Work illuminates the interplay between technical and academic mathematics. This collection of thought-provoking essays—by mathematicians, educators, and other experts—is enhanced with illustrative tasks from workplace and everyday contexts that suggest ways to strengthen high school mathematical education.

This important book addresses how to make mathematical education of all students meaningful—how to meet the practical needs of students entering the work force after high school as well as the needs of students going on to postsecondary education.

The short readable essays frame basic issues, provide background, and suggest alternatives to the traditional separation between technical and academic mathematics. They are accompanied by intriguing multipart problems that illustrate how deep mathematics functions in everyday settings—from analysis of ambulance response times to energy utilization, from buying a used car to "rounding off" to simplify problems.

The book addresses the role of standards in mathematics education, discussing issues such as finding common ground between science and mathematics education standards, improving the articulation from school to work, and comparing SAT results across settings.

Experts discuss how to develop curricula so that students learn to solve problems they are likely to encounter in life—while also providing them with approaches to unfamiliar problems. The book also addresses how teachers can help prepare students for postsecondary education.

For teacher education the book explores the changing nature of pedagogy and new approaches to teacher development. What kind of teaching will allow mathematics to be a guide rather than a gatekeeper to many career paths? Essays discuss pedagogical implication in problem-centered teaching, the role of complex mathematical tasks in teacher education, and the idea of making open-ended tasks—and the student work they elicit—central to professional discourse.

High School Mathematics at Work presents thoughtful views from experts. It identifies rich possibilities for teaching mathematics and preparing students for the technological challenges of the future. This book will inform and inspire teachers, teacher educators, curriculum developers, and others involved in improving mathematics education and the capabilities of tomorrow's work force.

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Essays and thoughts on mathematics

Many distinguished mathematicians, at some point of their career, collected their thoughts on mathematics (its aesthetic, purposes, methods, etc .) and on the work of a mathematician in written form.

For instance:

• W. Thurston wrote the lovely essay On proof and progress in mathematics in response to an article by Jaffe and Quinn ; some points made there are also presented in an answer given on MathOverflow ( What's a mathematician to do? ).
• More recently, T. Tao shared some personal thoughts and opinions on what makes "good quality mathematics" in What is good mathematics? .
• G. Hardy wrote the famous little book A Mathematician's Apology , which influenced, at least to some extent, several generations of mathematicians.

Personally, I've been greatly inspired by the two writings listed under (1.) -- they are one of the main reasons why I started studying mathematics -- and, considering that one of them appeared on MathOverflow , I'd like to propose here -- if it is appropriate -- to create a " big-list " of the kind of works described in the above blockquote.

I'd suggest (again, if it is appropriate) to give one title (or link) per answer with a short summary.

• A related question, which I've found very interesting, is Good papers/books/essays about the thought process behind mathematical research .
• Only slightly related (but surely interesting): Which mathematicians have influenced you the most?
• A single paper everyone should read? is not quite related, but still somewhat relevant (especially the most up-voted answer).
• reference-request
• soft-question
• 1 $\begingroup$ Hardy's apology is available here: math.ualberta.ca/~mss/misc/A%20Mathematician%27s%20Apology.pdf $\endgroup$ –  Goldstern Commented Oct 5, 2015 at 15:33
• $\begingroup$ This seems a little broad--can you be a bit more specific? I gave one answer, but do you want things like Dyson's "Birds and Frogs" or Gower's "Two cultures"? $\endgroup$ –  Kimball Commented Oct 5, 2015 at 22:29
• $\begingroup$ @Kimball, first of all, thanks for your answer, the book you suggested seems very interesting. Then, yes, I've read both those articles and, although they didn't come to my mind when I asked the question, they are surely two very insightful additions to this list. Thanks again. :) $\endgroup$ –  user81051 Commented Oct 6, 2015 at 18:12

22 Answers 22

There are many snippets that can be found. I like the following bit of the foreword by Thurston to J. H. Hubbard's Teichmüller Theory . I share the remarks because I think you simply can't have enough of Bill Thurston's insights:

"Mathematics is a paradoxical, elusive subject, with the habit of appearing clear and straightforward, then zooming away and leaving us stranded in a blank haze. Why? It is easy to forget that mathematics is primarily a tool for human thought. Mathematical thought is far better defined and far more logical than everyday thought, and people can be fooled into thinking of mathematics as logical, formal, symbolic reasoning. But this is far from reality. Logic, formalization, and symbols can be very powerful tools for humans to use, but we are actually very poor at purely formal reasoning; computers are far better at formal computation and formal reasoning, but humans are far better mathematicians. The most important thing about mathematics is how it resides in the human brain. Mathematics is not something we sense directly: it lives in our imagination and we sense it only indirectly. The choices of how it flows in our brains are not standard and automatic, and can be very sensitive to cues and context. Our minds depend on many interconnected special-purpose but powerful modules. We allocate everyday tasks to these various modules instinctively and subconsciously. The term `geometry', for instance, refers to a pattern of processing within our brains related to our spatial and visual senses, more than it refers to a separate content area of mathematics. One illustration of this is the concept of correlation between two measurements on a set, which is formally nearly identical with the concept of cosine of the angle between two vectors. The content is almost the same (for correlation, you first project to a hyperplane before measuring the cosine of the angle), but the human psychology is very different. Each mode of thinking has its own power, and ideally, people harness both modes of thought to work together. However, in formalized expositions, this psychological > difference vanishes. In the same way, any idea in mathematics can be thought about in many different ways, with competing advantages. When mathematics is explained, formalized and written down, there is a strong tendency to favor symbolic modes of thought at the expense of everything else, because symbols are easier to write and more standardized than other modes of reasoning. But when mathematics loses its connection to our minds, it dissolves into a haze. I've loved to read all my life. I went to New College of Sarasota, Florida, a small college that was just starting up with a strong emphasis on independent study, so I ended up learning a good deal of mathematics by reading mathematics books. At that time, I prided myself in reading quickly. I was really amazed by my first encounters with serious mathematics textbooks. I was very interested and impressed by the quality of the reasoning, but it was quite hard to stay alert and focused. After a few experiences of reading a few pages only to discover that I really had no idea what I'd just read, I learned to drink lots of coffee, slow way down, and accept that I needed to read these books at 1/10th or 1/50th standard reading speed, pay attention to every single word and backtrack to look up all the obscure numbers of equations and theorems in order to follow the arguments. Even so, when something was ``left to the reader'', I generally left it as well. At the time, I could appreciate that the mathematics was an impressive intellectual edifice, and I could follow the steps of proofs. I assumed that such an elaborate buildup must be leading to a fantastic denouement, which I eagerly awaited -- and waited, and waited. It was only much later, after much of the mathematics I had studied had come alive for me that I came to appreciate how ineffective and denatured the standard ((definition theorem proof)^n remark)^m style is for communicating mathematics. When I reread some of these early texts, I was stunned by how well their formalism and indirection hid the motivation, the intuition and the multiple ways to think about their subjects: they were unwelcoming to the full human mind. John Hubbard approaches mathematics with his whole mind. If you page through the current book, you will see many intriguing figures. That is a first sign: figures are one of the most important ways to keep our thought processes going in our whole brains, rather than settling down into the linguistic, symbol-handling areas. Of course, the figures in your imagination are even more important. Geometric ideas can be conveyed with words and with symbols, sometimes more effectively than with pictures, but a lack of figures is a good indication of a lack of geometry. Another important part of human thinking is the emotional aspect. In mathematics, what is intriguing, puzzling, interesting, surprising, boring, tedious, exciting is crucial; they are not incidental, they shape how we think. Personally, my thinking was shaped by boredom: I develop intense urges to come up with `easy' methods in order to avoid tedious computations that are opaque to me. Hubbard, a principal participant in the mathematics he is discussing, has done an excellent job in conveying the drama."

There are also many very good interviews that can be found, such as this one with Connes , as well as the advice to young mathematicians in the Princeton Companion to Mathematics .

A Mathematician's lament by Paul Lockhart: Reflections on how badly mathematics are taught these days. Imagining how it would be if music was taught the same way.

Indiscrete Thoughts by Gian-Carlo Rota and Discrete Thoughts by Kac, Rota, and Schwartz.

Proofs and Refutations: The Logic of Mathematical Discovery by Imre Lakatos: The sequence of steps through which mathematical ideas can be made to grow in an informal setting is explained through Socratic dialogues between a teacher and students. A beautiful read.

Since you mentioned A Mathematician's Apology : Michael Harris' Mathematics Without Apology .

Here's an excerpt explaining the title:

These attempts at justifications are the 'apologies' of the title. They usually take one of three forms. Pure research in mathematics as in other fields is good because it often leads to useful consequences (Steven Shapin calls this the Golden Goose argument); it is true because it offers a privileged access to certain truths; it is beautiful , an art form. To claim that these virtues are present in mathematics is not wrong, but it sheds little light on what is distinctively mathematical and even less about pure mathematicians' intentions . Intentions lie at the core of this book. I want to give the reader a sense of the mathematical life -- what it feels like to be a mathematician in a society of mathematicians where the first and second lives overlap.

Love and Math: The Heart of Hidden Reality by Edward Frenkel is, in my opinion, a lot better than Lockhart's lament.

The Mathematical Experience by Philip J. Davis and Reuben Hersh is a wonderful collection of essays on mathematics and on the experiences and culture of mathematicians. Written back in the 1980's, it has extremely insightful discussions of many of the same topics that nowadays are discussed on MO. For example, the essay "The Ideal Mathematician," which describes a hypothetical "ideal" mathematician working on the made-up area of "non-Riemannian hypersquares" is absolutely hilarious. Highly recommended!

• 1 $\begingroup$ The "Ideal Mathematician" is, to my mind, a poor mathematician. (It was a caricature, yes, but one which was a little too extreme for me.) $\endgroup$ –  Todd Trimble Commented Oct 5, 2015 at 16:29
• 1 $\begingroup$ @ToddTrimble, I disliked it too. For myself, the more bearing what I'm working on has on undergraduate or even high-school mathematics, the more excited I am about it. $\endgroup$ –  goblin GONE Commented Aug 23, 2016 at 14:55

Mathematics as Metaphor by Yuri Manin (both the title of the linked book which is a collection of essays, as well as the title of one particular essay in there). At least some of the essays you can find online.

I Want to be a Mathematician , by Paul Halmos.

• $\begingroup$ Indeed I love that book. Thanks for adding it. $\endgroup$ –  user81051 Commented Oct 6, 2015 at 18:13

Eugene Wigner: The Unreasonable Effectiveness of Mathematics in the Natural Sciences

The statement that the laws of nature are written in the language of mathematics was probably made three hundred years ago [It is attributed to Galileo]. It is now more true than ever before … Surely complex numbers are far from natural or simple and they cannot be suggested by physical observations. Furthermore, the use of complex numbers is close to being a necessity in the formulation of the laws of quantum mechanics. It is difficult to avoid the impression that a miracle confronts us here , quite comparable in its striking nature to the miracle that the human mind can string a thousand arguments together without getting itself into contradictions, or to the two miracles of the existence of laws of nature and of the human mind’s capacity to divine them. The closest explanation [for this mathematical universe] is Einstein’s statement that “the only physical theories which we are willing to accept are the beautiful ones” … the concepts of mathematics have this quality of beauty.

• 2 $\begingroup$ I have to disagree. Wigner's assertion that "mathematics is the science of skillful operations with concepts and rules invented just for this purpose" is the whole basis of his piece, and it doesn't have much to do with mathematics. The article is quasi-religious speculation based on this false premise. (The example that Wigner opens the article with is a case in point - he marvels at the appearance of $\sqrt{\pi}$ in the pdf for the normal distribution, as if this were magic. But probability theory was developed with very practical applications in mind.) $\endgroup$ –  Paul Levy Commented May 23, 2017 at 9:31

A Drifter of Dadaist Persuasion by Matilde Marcolli, published in Art in the Life of Mathematicians (Edited by Anna Kepes Szemerédi) American Mathematical Society, 2015, pp.210-231

The Psychology of Invention in the Mathematical Field (Jacques Hadamard's 1945 essay)

• $\begingroup$ This book was very influential to me, and made a huge difference in helping me understand m own process of doing mathematics. $\endgroup$ –  Zach H Commented Jul 17, 2017 at 17:13
• $\begingroup$ I love "the Poincare-Hadamard metaphor" described there! It says that our thoughts conscious and unconscious ones and their interactions could be explained via a mechanical model of states of a system of particles(the details inside). Very inspiring and still I haven't found an enough obstruction to the presented point of view there to the modern neuroscience, but I do not know much about it. An expertise needed! :) $\endgroup$ –  P. Grabowski Commented Apr 14, 2020 at 18:42

The Mathematician by John Von Neumannn.

Enigmas of Chance , by Mark Kac.

I would add "Letters to a Young Mathematician" by Ian Stewart

I recommend:

Vladimir Arnold: Yesterday and Long Ago . This is a very enjoyable and highly interesting collection of anecdotes and historical remarks. The latest Russian edition of this book contains some more chapters. Richard Hamming: You and Your Research , transcribed and edited by J F Kaiser, reprinted in Tveito et al: Simula Research Laboratory . This is the text of a lecture of Hamming.

Birth of a Theorem , by French candidate for Parliament Cédric Villani

• 4 $\begingroup$ Now French member of Parliament Cédric Villani. $\endgroup$ –  Michael Lugo Commented Jul 17, 2017 at 15:16

Here are additional mathematicians' thoughts.

S. Ulam, Adventures of a mathematician .A recollection of his life, from Lwow to Los Alamos. I am linking to excerpts. The book is still available for purchase.

Advices to a Young mathematician , a collection of advice and anecdotes by M. Atiyah, B. Bollobas, A. Connes, D. McDuff and P. Sarnak.

A. Borel, Art and science (Math. Intelligencer vol.5 1983, translation from German). A text for a general audience about the relationship between art and mathematics.

R. P. Langlands Is there beauty in mathematical theories? , this text is actually about number theory, old and new.

T. Gowers The two cultures of mathematics , another take on the dichotomy between problem solving and theory building.

A. Connes A view of mathematics , a thorough exposition of A. Connes'philosophical stance about space and physics. Targeted at a scientific audience.

D. Mumford, the dawning of the age of stochasticity , from algebraic geometry to statistics.

Y. Manin, Interrelations between Mathematics and Physics , on the divergence between mathematics and physics in the XXe century.

M. Gromov, ergobrain , one of the most surprising inquiry about life and mathematics.

I end that list with a text from a french mathematician about the future of mathematics: Poincare, l'avenir des mathematiques .

Perhaps a little broader in range/scope than the original question intended — but then again, perhaps not — the essays collected in

Mathématiques, mathematiciens et société. Publications Mathématiques d'Orsay no. 86 74-16 (1974)

I was led to this when someone somewhere posted a link to Vergne's Témoignage d'une mathématicienne , which is one of the essays in this volume, and — I must confess — is the only one I've read, although the other ones do look interesting

In the Princeton Companion to Mathematics , there is a section entitled Advice to a Young Mathematician (pdf), containing essays by Atiyah, Bollobás, Connes, McDuff and Sarnak.

A Mathematician's Miscellany (reprinted, with additional material, as Littlewood's Miscellany by CUP in 1986) is worthwhile reading.

Clifford Truesdell published a series of essays as An Idiot's Fugitive Essays on Science Methods, Criticism, Training, Circumstances (Springer, 1984), which sets out in a forthright manner the author's views on mathematics and science.

A really nice article by Andrei Toom about mathematical education, especially in the US, got recently mentioned in a comment to this question.

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Math Essay Ideas for Students: Exploring Mathematical Concepts

Are you a student who's been tasked with writing a math essay? Don't fret! While math may seem like an abstract and daunting subject, it's actually full of fascinating concepts waiting to be explored. In this article, we'll delve into some exciting math essay ideas that will not only pique your interest but also impress your teachers. So grab your pens and calculators, and let's dive into the world of mathematics!

• The Beauty of Fibonacci Sequence

Have you ever wondered why sunflowers, pinecones, and even galaxies exhibit a mesmerizing spiral pattern? It's all thanks to the Fibonacci sequence! Explore the origin, properties, and real-world applications of this remarkable mathematical sequence. Discuss how it manifests in nature, art, and even financial markets. Unveil the hidden beauty behind these numbers and show how they shape the world around us.

• The Mathematics of Music

Did you know that music and mathematics go hand in hand? Dive into the relationship between these two seemingly unrelated fields and develop your writing skills . Explore the connection between harmonics, frequencies, and mathematical ratios. Analyze how musical scales are constructed and why certain combinations of notes create pleasant melodies while others may sound dissonant. Explore the fascinating world where numbers and melodies intertwine.

• The Geometry of Architecture

Architects have been using mathematical principles for centuries to create awe-inspiring structures. Explore the geometric concepts that underpin iconic architectural designs. From the symmetry of the Parthenon to the intricate tessellations in Islamic art, mathematics plays a crucial role in creating visually stunning buildings. Discuss the mathematical principles architects employ and how they enhance the functionality and aesthetics of their designs.

• Fractals: Nature's Infinite Complexity

Step into the mesmerizing world of fractals, where infinite complexity arises from simple patterns. Did you know that the famous Mandelbrot set , a classic example of a fractal, has been studied extensively and generated using computers? In fact, it is estimated that the Mandelbrot set requires billions of calculations to generate just a single image! This showcases the computational power and mathematical precision involved in exploring the beauty of fractal geometry.

Explore the beauty and intricacy of fractal geometry, from the famous Mandelbrot set to the Sierpinski triangle. Discuss the self-similarity and infinite iteration that define fractals and how they can be found in natural phenomena such as coastlines, clouds, and even in the structure of our lungs. Examine how fractal mathematics is applied in computer graphics, art, and the study of chaotic systems. Let the captivating world of fractals unfold before your eyes.

• The Game Theory Revolution

Game theory isn't just about playing games; it's a powerful tool used in various fields, from economics to biology. Dive into the world of strategic decision-making and explore how game theory helps us understand human behavior and predict outcomes. Discuss in your essay classic games like The Prisoner's Dilemma and examine how mathematical models can shed light on complex social interactions. Explore the cutting-edge applications of game theory in diverse fields, such as cybersecurity and evolutionary biology. If you still have difficulties choosing an idea for a math essay, find a reliable expert online. Ask them to write me an essay or provide any other academic assistance with your math assignments.

• Chaos Theory and the Butterfly Effect

While writing an essay, explore the fascinating world of chaos theory and how small changes can lead to big consequences. Discuss the famous Butterfly Effect and how it exemplifies the sensitive dependence on initial conditions. Delve into the mathematical principles behind chaotic systems and their applications in weather forecasting, population dynamics, and cryptography. Unravel the hidden order within apparent randomness and showcase the far-reaching implications of chaos theory.

• The Mathematics Behind Cryptography

In an increasingly digital world, cryptography plays a vital role in ensuring secure communication and data protection. Did you know that the global cybersecurity market is projected to reach a staggering $248.26 billion by 2023? This statistic emphasizes the growing importance of cryptography in safeguarding sensitive information.

Explore the mathematical foundations of cryptography and how it allows for the creation of unbreakable codes and encryption algorithms. Discuss the concepts of prime numbers, modular arithmetic, and public-key cryptography. Delve into the fascinating history of cryptography, from ancient times to modern-day encryption methods. In your essay, highlight the importance of mathematics in safeguarding sensitive information and the ongoing challenges faced by cryptographers.

Writing a math essay doesn't have to be a daunting task. By choosing a captivating topic and exploring the various mathematical concepts, you can turn your essay into a fascinating journey of discovery. Whether you're uncovering the beauty of the Fibonacci sequence, exploring the mathematical underpinnings of music, or delving into the game theory revolution, there's a world of possibilities waiting to be explored. So embrace the power of mathematics and let your creativity shine through your words!

Remember, these are just a few math essay ideas to get you started. Feel free to explore other mathematical concepts that ignite your curiosity. The world of mathematics is vast, and each concept has its own unique story to tell. So go ahead, unleash your inner mathematician, and embark on an exciting journey through the captivating realm of mathematical ideas!

Tobi Columb, a math expert, is a dedicated educator and explorer. He is deeply fascinated by the infinite possibilities of mathematics. Tobi's mission is to equip his students with the tools needed to excel in the realm of numbers. He also advocates for the benefits of a gluten-free lifestyle for students and people of all ages. Join Tobi on his transformative journey of mathematical mastery and holistic well-being.

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Who Uses Maths? Almost Everyone!

In the midst of Maths Week Ireland , many students may be asking “W hat use is mathematics and what purpose is served by studying it?” Mathematicians often stress the inherent beauty and intellectual charm of the subject, but that is unlikely to persuade many people, who demand to know how mathematics can be of use and value to them. [ TM149 , or search for “thatsmaths” at  irishtimes.com ].

In reality, mathematics is essential in numerous contexts: t he diversity is remarkable, and you may be surprised how maths plays a vital role in the everyday work of so many people.

Who uses mathematics in their work? The list is very long, but a sample might include accountants, actuaries, air traffic controllers, animators, architects, astronomers, builders, cartographers, chemists, cryptographers, economists, electricians, engineers, financiers, farmers, geologists, hydrologists, journalists, pharmacists, photographers, pilots, plumbers, stockbrokers, surveyors, teachers, urban planners and web designers. Many more careers might be added to this list.

“ We Use Maths”

Students who have difficulties learning mathe matics may wonder “what’s the point?” The website weusemaths.ie has been constructed by Colin Hogg of the Dublin Institute for Advanced Studies (DIAS) with the aim of showing how maths is vital for many great careers. This website makes connections between the mathematics curriculum for the Junior and Leaving Certificate and the wide range of people who use maths every day in the course of their work. The key idea is to make clear links between the curriculum, courses and careers.

School Curriculum

The school mathematics curriculum was designed to enable students to develop the problem-solving skills that are needed in higher education and in the workplace. The idea is to study mathematics in the context of challenging, real-world problems. The five strands of the new curriculum are Statistics and Probability, Geometry and Trigonometry, Number, Algebra and Functions. An outline of the curriculum is given on weusemaths.ie , and also a link to the official NCCA website for full details.

Statistics and probability are perhaps the most commonly applied areas of mathematics, used by journalists, stockbrokers, nurses, athletes, data scientists, actuaries and, of course, statisticians. Geometry and trigonometry are crucial for engineers, surveyors, architects, geographers and web designers. Algebra is used by market analysts, animators, software developers and geoscientists. Functions and calculus are essential for meteorologists, engineers, pharmaceutical scientists, hydrologists and everyone involved in mathematical modelling. The topics studied under “Numbers” are of use to essentially everybody, at work, at home and at play.

Under the heading “Courses & Careers”, the website gives information about 50 different careers, the mathematics involved and the pathways by which students may enter each career. Students who know what career they would like to pursue, but who don’t know how to go about it, may find this a very valuable resource. All the occupations in the long list above are included, and many more besides. The “We Use Maths” site also includes a list of partners involved in developing the site and a link to contact the website to pose questions or provide feedback.

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The Best Writing on Mathematics 12

Mircea pitici,  series editor.

This annual anthology brings together the year’s finest mathematics writing from around the world. Featuring promising new voices alongside some of the foremost names in the field, The Best Writing on Mathematics makes mathematical writing available to a wide audience.

The year’s finest mathematical writing from around the world

The year's finest mathematical writing from around the world

The year's finest mathematics writing from around the world

The year's finest writing on mathematics from around the world

The year's finest writing on mathematics from around the world, with a foreword by Nobel Prize – winning physicist Roger Penrose

The year’s most memorable writing on mathematics

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

• Conference paper
• Open Access
• First Online: 02 November 2017
• Cite this conference paper

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• Günter M. Ziegler 3 &
• Andreas Loos 4  

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

112k Accesses

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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.

You have full access to this open access chapter,  Download conference paper PDF

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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.

Blum, W. (2001). Was ist was. Band 12: Mathematik , Tessloff Verlag, Nürnberg. Revised version, with new cover, 2010.

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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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