essays on mathematics

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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 | Essay on Math for Students and Children in English

February 13, 2024 by Prasanna

Math Essay: Mathematics is generally defined as the science that deals with numbers. It involves operations among numbers, and it also helps you to calculate the product price, how many discounted prizes here, and If you good in maths so you can calculate very fast. Mathematicians and scientists rely on mathematics principles in their real-life to experiments with new things every day. Many students say that ” I hate mathematics ” and maths is a useless subject, but it is wrong because without mathematics your life is tough to survive. Math has its applications in every field.

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

Long and Short Essays on Math for Students and Kids in English

We are presenting students with essay samples on an extended essay of 500 words and a short of 150 words on the topic of math for reference.

Long Essay on Math 500 Words in English

Long Essay on Math is usually given to classes 7, 8, 9, and 10.

Mathematics is one of the common subjects that we study since our childhood. It is generally used in our daily life. Every person needs to learn some basics of it. Even counting money also includes math. Every work is linked with math in some way or the other. A person who does math is called a Mathematician.

Mathematics can be divided into two parts. The first is Pure mathematics, and the second is Applied mathematics. In Pure mathematics, we need to study the basic concept and structures of mathematics. But, on the other side, Applied mathematics involves the application of mathematics to solve problems that arise in various areas,(e.g.), science, engineering, and so on.

One couldn’t imagine the world without math. Math makes our life systematic, and every invention involves math. No matter what action a person is doing, he should know some basic maths. Every profession involves maths. Our present-day world runs on computers, and even computer runs with the help of maths. Every development that happens requires math.

Mathematics has a wide range of applications in our daily life. Maths generally deals with numbers. There are various topics in math, such as trigonometry; integration; differentiation, etc. All the subjects such as physics; chemistry; economy; commerce involve maths in some way or the other. Math is also used to find the relation between two numbers, and math is considered to be one of the most challenging subjects to learn. Math includes various numbers, and many symbols are used to show the relation between two different numbers.

Math is complicated to learn, and one needs to focus and concentrate more. Math is logical sometimes, and the logic needs to be derived out. Maths make our life easier and more straightforward. Math is considered to be challenging because it consists of many formulas that have to be learned, and many symbols and each symbol generally has its significance.

Some of the advantages of Math in our daily life

• Managing Money: Counting money and calculating simple interest, compound interest includes the usage of mathematics. Profit and loss are also computed using maths. Anything related to maths contains maths.
• Cooking: Maths is even used in cooking as estimating the number of ingredients that have to be used is calculated in numbers. Proportions also include maths.
• Home modelling: Calculating the area is essential in the construction of the home or home modelling. The size is also measured using maths. Even heights are also measured using maths.
• Travelling: Distance between two places and time taken to travel also includes maths. The amount of time taken revolves around maths. Almost every work is related to maths in some way or another. Maths contains some conditions that need to be followed, and maths has several formulas that have to be learned to become a mathematician.

Short Essay on Math 150 Words in English

Short Essay on Math is usually given to classes 1, 2, 3, 4, 5, and 6.

Maths is generally defined as the science of numbers and the operations performed among them. It deals with both alphabets along with numbers and involves addition, subtraction, multiplication, division, comparison, etc. It is used in every field. Maths consists of finding a relation between numbers, calculating the distance between two places, counting money, calculating profit and loss.

It is of two types pure and applied. Pure math deals with the basic structure and concept of maths, whereas applied mathematics deals with how maths is used it involves the application of maths in our daily life. All the subjects include maths, and hence maths is considered to be one of the primary and joint issues which need to be learned by everyone. One couldn’t imagine their life using maths. It has made our experience easy and straightforward. It has prevented chaos in our daily life. Hence learning maths is mandatory for everyone.

10 Lines on Math in English

• Father of Mathematics was Archimedes.
• Hypatia is the first woman know to know to have taught mathematics.
• From 0-1000 ,letter “A” only appears first in 1,000 ( “one thousand “).
• Zero (0) is the only number that can not be represented by Roman numerals.
• The Sign plus (+) and Minus(-) were discovered in 1489 A.D.
• Do you know that a Baseball field is of the perfect shape of a Rhombus.
• Jiffy is considered to be a unit of time for 1/100th of a second.
• 14th March International Day of Mathematics.
• Most mathematics symbols weren’t invented until the 16th century.
• The symbols for the division is called an Obelus.

FAQ’s on Math Essay

Question 1. What is Mathematics in simple words?

Answer: Mathematics is the study of shapes, patterns, numbers, and more. It involves a comparison between two numbers and calculating the distance between two places.

Question 2. Do we need mathematics every day?

Answer: Yes, we need mathematics every day, from buying a product to sell anything you want. Maths is present in our daily life, and no matter what work we do, maths is involved, and the application of maths is current in our everyday life.

Question 3. Who was the No.1 Mathematicians in the world?

Answer: Isaac Newton, who was a profound mathematician, is considered to be one of the best mathematicians in the world.

Question 4. What are the applications of maths?

Answer: Maths have various applications in our daily life. Maths is present everywhere from counting money to the calculating distance between two places. We could find math applications around.

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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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• Mathematics in Sports: Analyzing the Numbers Behind Athletic Performance – This essay would explore the use of mathematics in sports, including the use of statistics and analytics to analyze athletic performance and predict outcomes.
• Chaos Theory: The Science of Nonlinear Systems – This essay would discuss the concept of chaos theory and its applications in various fields, such as meteorology, physics, and economics. It would explore the idea that small changes in initial conditions can have a significant impact on the final outcome of a system.
• The Mathematics of Cryptography: Securing Information in the Digital Age – This essay would examine the use of mathematics in cryptography, including the principles of encryption and decryption, and how these concepts are applied to secure information in the digital age.
• Fractals: The Beauty of Infinite Complexity – This essay would explore the concept of fractals and their applications in art, nature, and science. It would discuss the beauty and complexity of these repeating patterns found in nature and how they are used in various fields of study.
• Mathematical Models in Biology: Understanding the Complexities of Life – This essay would discuss the use of mathematical models in biology, including the modeling of population growth, the spread of disease, and the behavior of organisms. It would explore how these models help scientists understand the complex systems that make up living organisms.
• The Mathematics of Finance: Analyzing Investments and Markets – This essay would examine the use of mathematics in finance, including the principles of financial analysis, investments, and risk management. It would explore how mathematics is used to understand and predict market trends.
• Geometry in Art: The Intersection of Math and Creativity – This essay would discuss the use of geometry in art, including the use of shapes, patterns, and symmetry. It would explore how artists use mathematical concepts to create beautiful and compelling works of art.
• The History of Mathematics: From Ancient Times to Modern-Day Advances – This essay would trace the history of mathematics, from its origins in ancient civilizations to modern-day advancements in the field. It would explore the contributions of key mathematicians throughout history and the evolution of mathematical concepts and principles over time.

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High School Mathematics at Work

Essays and examples for the education of all students.

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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Mathematics

The year’s finest mathematical writing from around the world

• The Best Writing on Mathematics

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This annual anthology brings together the year’s finest mathematics writing from around the world—and you don’t need to be a mathematician to enjoy the pieces collected here. These essays—from leading names and fresh new voices—delve into the history, philosophy, teaching, and everyday aspects of math, offering surprising insights into its nature, meaning, and practice, and taking readers behind the scenes of today’s hottest mathematical debates. Here, Viktor Blåsjö gives a brief history of “lockdown mathematics”; Yelda Nasifoglu decodes the politics of a seventeenth-century play in which the characters are geometric shapes; and Andrew Lewis-Pye explains the basic algorithmic rules and computational procedures behind cryptocurrencies. In other essays, Terence Tao candidly recalls the adventures and misadventures of growing up to become a leading mathematician; Natalie Wolchover shows how old math gives new clues about whether time really flows; and David Hand discusses the problem of “dark data”—information that is missing or ignored. And there is much, much more.

Praise for previous editions:“A variety of thoroughly accessible works that tie abstract math to the real world. . . . Gives readers an entertaining look at the odd, the amusing, and the utilitarian without requiring any more than a readerly curiosity.” —Publishers Weekly “Wonderful. . . . Cannot be recommended highly enough!”—Robert Schaefer, New York Journal of Books “A wonderful and varied bouquet of texts. . . . I highly recommend this book.”—Stephen Buckley, Irish Mathematical Society Bulletin

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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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Discrete Thoughts: Essays on Mathematics, Science and Philosophy 2nd Edition

• ISBN-10 0817636366
• ISBN-13 978-0817636364
• Edition 2nd
• Publisher Birkhäuser
• Publication date June 1, 1993
• Language English
• Dimensions 6.1 x 0.64 x 9.25 inches
• Print length 278 pages
• See all details

Editorial Reviews

From the reviews of the second edition:

"...These papers reflect on mathematics and its influence on human society. They can help the specialist to notice what is going on around him, and they may lead educated people from other domains to a better understanding of mathematics. Many of these papers can advise educators how to form a modern mathematics education, which develops approved ideas and institutions...I admire the stimulating perspectives of the authors"

--American Mathematical Society

"Kac, Rota, and Schwartz have collected here: occasional essays about mathematics, mathematicians, and surrounding subjects. … The essays are fun to read, light in manner but serious in content. Many of them are provocative. … As a result, the book would make wonderful fodder for a reading course or seminar. … Discrete Thoughts is a classic indeed." (Fernando Q. Gouvêa, MathDL, January, 2008)

From the Back Cover

This is a volume of essays and reviews that delightfully explore mathematics in all its moods ― from the light and the witty, and humorous to serious, rational, and cerebral. Topics include: logic, combinatorics, statistics, economics, artificial intelligence, computer science, and applications of mathematics broadly. You will also find history and philosophy covered, including discussion of the work of Ulam, Kant, Heidegger among others.

"...these papers reflect on mathematics and its influence on human society. They can help the specialist to notice what is going on around him, and they may lead educated people from other domains to a better understanding of mathematics. Many of these papers can advise educators how to form a modern mathematics education, which develops approved ideas and institutions...I admire the stimulating perspectives of the authors."---American Mathematical Society

"‘Mathematicians, like Proust and everyone else, are at their best when writing about their first love’ … They are among the very best we have; and their best is very good indeed. … One approaches this book with high hopes. Happily, one is not disappointed."--- From The Mathematical Intelligencer

"Mathematics is shaped by the consistent concerns and styles of powerful minds ―three of which are represented here. Kac’s work is marked by deep commitment and breadth of inquiry. Rota is the easiest of these authors to read…a delight: witty and urbane, with a clear and interesting agenda and an astonishing intellectual range. To read him is to be a part of a pleasant and rewarding conversation.―Mathematics Magazine

Product details

• Publisher ‏ : ‎ Birkhäuser; 2nd edition (June 1, 1993)
• Language ‏ : ‎ English
• Paperback ‏ : ‎ 278 pages
• ISBN-10 ‏ : ‎ 0817636366
• ISBN-13 ‏ : ‎ 978-0817636364
• Item Weight ‏ : ‎ 14.4 ounces
• Dimensions ‏ : ‎ 6.1 x 0.64 x 9.25 inches
• #727 in Game Theory (Books)
• #1,657 in Mathematical Logic
• #2,047 in Mathematics History

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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.

Thus, math plays a vital role in our daily lives, making it an essential subject to learn.

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.

That’s it! I hope the essay helped you.

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Discrete Thoughts

Essays on Mathematics, Science and Philosophy

• © 1992
• Latest edition
• Gian-Carlo Rota 1 ,
• Jacob T. Schwartz 2

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Dept. of Mathematics and Philosophy, MIT, Cambridge, USA

Courant institute, new york, usa.

• Beautifully written articles from three great modern mathematicians
• Ideas provoke thought and debate
• Provides a source for supplemental reading for almost any math class

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What Is Mathematics and What Should It Be?

Introduction

Do mathematicians have responsibilities.

• History of Mathematics
• mathematics

Table of contents (26 chapters)

Front matter.

• Mark Kac, Gian-Carlo Rota, Jacob T. Schwartz

Mathematics: Tensions

The pernicious influence of mathematics on science, statistics and its history, combinatorics, computer science, mathematics: trends, the future of computer science, economics, mathematical and empirical, complicating mathematics, mathematics and its history, academic responsibility, husserl and the reform of logic, artificial intelligence, computing and its history, authors and affiliations.

Gian-Carlo Rota

Jacob T. Schwartz

Bibliographic Information

Book Title : Discrete Thoughts

Book Subtitle : Essays on Mathematics, Science and Philosophy

Authors : Mark Kac, Gian-Carlo Rota, Jacob T. Schwartz

DOI : https://doi.org/10.1007/978-0-8176-4775-9

Publisher : Birkhäuser Boston, MA

eBook Packages : Springer Book Archive

Copyright Information : Springer Science+Business Media New York 1992

Softcover ISBN : 978-0-8176-3636-4 Published: 01 June 1993

eBook ISBN : 978-0-8176-4775-9 Published: 01 July 2009

Edition Number : 2

Number of Pages : XII, 266

Additional Information : Originally published as a monograph

Topics : History of Mathematical Sciences , Mathematical Logic and Foundations , Applications of Mathematics , Game Theory, Economics, Social and Behav. Sciences , Mathematics, general

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Extended Essay: Group 5: Mathematics

• General Timeline
• Group 1: English Language and Literature
• Group 2: Language Acquisition
• Group 3: Individuals and Societies
• Group 4: Sciences
• Group 5: Mathematics
• Group 6: The Arts
• Interdisciplinary essays
• Six sub-categories for WSEE
• IB Interdisciplinary EE Assessment Guide
• Brainstorming
• Pre-Writing
• Research Techniques
• The Research Question
• Paraphrasing, Summarising and Quotations
• Writing an EE Introduction
• Writing the main body of your EE
• Writing your EE Conclusion
• Sources: Finding, Organising and Evaluating Them
• Conducting Interviews and Surveys
• Citing and Referencing
• Check-in Sessions
• First Formal Reflection
• Second Formal Reflection
• Final Reflection (Viva Voce)
• Researcher's Reflection Space (RRS) Examples
• Information for Supervisors
• How is the EE Graded?
• EE Online Resources
• Stavanger Public Library
• Exemplar Essays
• Extended Essay Presentations
• ISS High School Academic Honesty Policy

Mathematics

An extended essay (EE) in mathematics is intended for students who are writing on any topic that has a mathematical focus and it need not be confined to the theory of mathematics itself.

Essays in this group are divided into six categories:

• the applicability of mathematics to solve both real and abstract problems
• the beauty of mathematics—eg geometry or fractal theory
• the elegance of mathematics in the proving of theorems—eg number theory
• the history of mathematics: the origin and subsequent development of a branch of mathematics over a period of time, measured in tens, hundreds or thousands of years
• the effect of technology on mathematics:
• in forging links between different branches of mathematics,
• or in bringing about a new branch of mathematics, or causing a particular branch to flourish.

These are just some of the many different ways that mathematics can be enjoyable or useful, or, as in many cases, both.

For an Introduction in a Mathematics EE look HERE . 

Choice of topic

The EE may be written on any topic that has a mathematical focus and it need not be confined to the theory of mathematics itself.

Students may choose mathematical topics from fields such as engineering, the sciences or the social sciences, as well as from mathematics itself.

Statistical analyses of experimental results taken from other subject areas are also acceptable, provided that they focus on the modeling process and discuss the limitations of the results; such essays should not include extensive non-mathematical detail.

A topic selected from the history of mathematics may also be appropriate, provided that a clear line of mathematical development is demonstrated. Concentration on the lives of, or personal rivalries between, mathematicians would be irrelevant and would not score highly on the assessment criteria.

It should be noted that the assessment criteria give credit for the nature of the investigation and for the extent that reasoned arguments are applied to an appropriate research question.

Students should avoid choosing a topic that gives rise to a trivial research question or one that is not sufficiently focused to allow appropriate treatment within the requirements of the EE.

Students will normally be expected either to extend their knowledge beyond that encountered in the Diploma Programme mathematics course they are studying or to apply techniques used in their mathematics course to modeling in an appropriately chosen topic.

However, it is very important to remember that it is an essay that is being written, not a research paper for a journal of advanced mathematics, and no result, however impressive, should be quoted without evidence of the student’s real understanding of it.

Example and Treatment of Topic

Examples of topics

These examples are just for guidance. Students must ensure their choice of topic is focused (left-hand column) rather than broad (right-hand column

Treatment of the topic

Whatever the title of the EE, students must apply good mathematical practice that is relevant to the

chosen topic, including:

• data analysed using appropriate techniques

• arguments correctly reasoned

• situations modeled using correct methodology

• problems clearly stated and techniques at the correct level of sophistication applied to their solution.

Research methods

Students must be advised that mathematical research is a long-term and open-ended exploration of a set of related mathematical problems that are based on personal observations. 

The answers to these problems connect to and build upon each other over time.

Students’ research should be guided by analysis of primary and secondary sources.

A primary source for research in mathematics involves:

• data-gathering

• visualization

• abstraction

• conjecturing

• proof.

A secondary source of research refers to a comprehensive review of scholarly work, including books, journal articles or essays in an edited collection.

A literature review for mathematics might not be as extensive as in other subjects, but students are expected to demonstrate their knowledge and understanding of the mathematics they are using in the context of the broader discipline, for example how the mathematics they are using has been applied before, or in a different area to the one they are investigating.

Writing the essay

Throughout the EE students should communicate mathematically:

• describing their way of thinking

• writing definitions and conjectures

• using symbols, theorems, graphs and diagrams

• justifying their conclusions.

There must be sufficient explanation and commentary throughout the essay to ensure that the reader does not lose sight of its purpose in a mass of mathematical symbols, formulae and analysis.

The unique disciplines of mathematics must be respected throughout. Relevant graphs and diagrams are often important and should be incorporated in the body of the essay, not relegated to an appendix.

However, lengthy printouts, tables of results and computer programs should not be allowed to interrupt the development of the essay, and should appear separately as footnotes or in an appendix. Proofs of key results may be included, but proofs of standard results should be either omitted or, if they illustrate an important point, included in an appendix.

Examples of topics, research questions and suggested approaches

Once students have identified their topic and written their research question, they can decide how to

research their answer. They may find it helpful to write a statement outlining their broad approach. These

examples are for guidance only.

An important note on “double-dipping”

Students must ensure that their EE does not duplicate other work they are submitting for the Diploma Programme. For example, students are not permitted to repeat any of the mathematics in their IA in their EE, or vice versa.

The mathematics EE and internal assessment

An EE in mathematics is not an extension of the internal assessment (IA) task. Students must ensure that they understand the differences between the two.

• The EE is a more substantial piece of work that requires formal research
• The IA is an exploration of an idea in mathematics.

It is not appropriate for a student to choose the same topic for an EE as the IA. There would be too much danger of duplication and it must therefore be discouraged.

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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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IB Maths EE examples

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The big myth that keeps people from loving math

• There is no secret code or single method to solve a mathematical problem.
• This “answer-getting” myth leads to disempowerment when learning math.
• To counteract the myth, don’t focus on the answer but the process.

Teaching math as if there’s only one correct way to solve a problem makes us think that we’re problem-solving, but instead, we’re “answer-getting.” I’ve seen it so many times, but none bothers me more than watching elementary and middle school students solve word problems in this way.

Consider this typical middle school prompt: A store is selling 6 bags of marbles for $18. What is the unit price for a bag of marbles? When I read this problem, I picture a child looking up at me and asking, “Does ‘of’ mean multiplication?” It has happened to me so many times when I visit math classrooms.

There’s no secret code. Of could mean “multiply,” but it might not. These are the highly counterproductive questions that children ask when they have been presented with a “single way” to solve word problems, such as looking for keywords. 

In this example, the students may immediately multiply 6 × 18. If you then ask them why the unit price of a single bag of marbles would cost $108 — and be so much more than the price of 6 bags of marbles — they will look at you with uncertainty. This is the end result of answer-getting.

Problem-solving is a distinct cognitive experience. Instead, we ask, what is happening in the problem? It is not mindlessly following a single prescriptive set of steps. The way to solve this problem, and every problem, is to understand what is happening. But that means there will be many paths to the answer. How I understand the problem might be quite unlike how you understand it.

The right way is the wrong way

When we are taught to rely on a singular, step-by-step process as the true way to solve a math problem, we turn off our problem-solving brain. These skills require continuous work to keep them sharp, and the constant reliance on someone else’s “exact right” method dulls them. Over the years, we may even lose at least some of our problem-solving acumen by not using it.

This reliance also discourages courage — we need to take chances to solve problems, and insistence on following a singular method prevents us from risking wrong answers via experimentation.

We can solve problems in many distinct ways. In fact, trying different approaches is fun as well as instructive, and it is necessary when problem-solving gets hard — which is often when the problems are most worth solving. Engineers who write software code or build bridges make a conscious attempt to solve problems in more than one way, even when a solution is readily available to them. 

Why not solve it and move on?

First of all, if you dig deeper to find more than one solution, you can decide which one among them is less expensive, more durable, or more elegant — whichever outcome matters most to you. Second, and perhaps more significant, when problem-solving gets really hard and the way ahead isn’t clear, you need to be ready to try anything. And the first step of the “try anything” approach is to back up and examine a problem from every angle, or at least from more angles than you initially see.

In the real world, of course, we often resort to looking at problems from fresh angles out of desperation. “Try anything” is the motto. As one member of a two-working-parents household with elementary school twins during the COVID-19 pandemic, we often were compelled to try anything to solve problems regarding work, social distancing protocols, on-and-off remote school, and limited childcare.

To expose the myth of a single correct method for the sham that it is, we need to understand the consequences of answer-getting versus problem-solving. Because we’ve been brainwashed into believing that answer-getting is good and because most of us spent years in answer-getting math curricula, we don’t realize the negative effects it has on us.

Here are typical ways we respond in an answer-getting environment:

• The mind goes blank. For a moment, nothing occurs to us because we’re not allowed to use our minds creatively.
• Racing heart. We react anxiously as we try to remember how the teacher did the math on the board. What was her first step again?
• Negative self-talk. For a moment, we have the germ of an idea, an instinct about how to start solving a challenging math problem, but because we’ve been conditioned to seek the answer only one way, we chastise ourselves for thinking we know better than what we’ve been taught, and we revert to standard operating procedure.
• Reluctance to talk through questions and concerns. We’re embarrassed to bring up these issues with others, assuming they are “right way” adherents. This reluctance to involve others is an obstacle to a creative, collaborative process.

The overarching effect of an answer-getting system is disempowerment. We feel defeated before even attempting to work on a problem.

Here are recent conversations I have had with children and adults on what this sort of math feels like:

“I want to use decimals. The teacher wants me to use fractions for no reason. I just have to do what he says. There is no freedom to do the math the way you want to do it, even if my way is easier for me. No one listens to me.”

“I actually remember getting dinged on a high school math test even when I had the right answer, but I had solved it my own way. As a teenager, that made me furious. Now looking back as an adult, I think about it like tennis. If you are drilling me so I learn or improve a new skill like backhand volleys, then I can understand the reasoning for forcing a specific approach. But if you have no reason whatsoever for forcing your way on me, it still steams me to think about it.”

A problem-solving approach conjures significantly contrasting responses — responses that reflect a sense of empowerment and courage. Ideally, schools would teach math with problem-solving as its driving principle rather than the myth of a single right way. To approach this ideal, however, we need to understand what problem-solving is all about.

Math should be taught as a collaborative process, much as other subjects are taught.

How to counteract the myth

We make math a performative rather than a learning experience. When the teacher asks the class, “What is the answer to 63 plus 37?” he turns math into an individual sport.

Add the myth of speed, and each student is scrambling to come up with the answer first and win the game. The answer becomes the only thing that matters, and both understanding and collaboration fall by the wayside.

No doubt, some of you might wonder if I’ve lost my math mind. After all, we need to get the answers right so that we can purchase the right amount of carpeting to cover a room’s floor or make sure that our rocket makes it to the moon. 

Again, this is an issue of integrative complexity. Of course, we need to know what 63 + 37 equals. But if that’s all we know, then we’re missing out on a lot of what math offers.

Fortunately, we can learn in a way that we obtain precision as well as other benefits. Consider again 63 + 37. What if the teacher framed the question this way: “Don’t tell me the answer. It’s 100. How would you start calculating 63 plus 37 in your head? What is your first step?” Now math is a process.

I have gotten the chance to hear second graders’ brains working at this moment many times. Each time is a joy. One might say, “I broke this up as 60 plus 30 plus 3 plus 7. And the next thing in my head I saw was that it was 93 plus 7. And then I knew that was 100.” Another second grader might offer another option: “I looked at this for a moment, and I saw that 3 and 7 make 10. So I knew I had a 60 plus 30 plus 10. And I know that’s 100.”

This is the math people need in their lives. This is what is needed to build bridges. It’s also how you build a deep math sense. 

Math should be taught as a collaborative process, much as other subjects are taught. We often view math as distinct from other subjects in K–8, as something that must be taught as an individual sport where everyone is on his or her own to come up with the right answer first. Other subjects are taught as team sports, ones where process matters, where students don’t rely on tricks, where students are encouraged to work together, and where a variety of ways to answer may be acceptable. But when it comes to math, collaboration and process work are subordinated or eliminated.

The Myth of the Math Kid

A cross the nation, kids are heading back to school. It’s an exciting time. I remember both the joy and the nervousness that came with my now twin 13-year-olds' first starting school. In fact, one day in particular stands out.

I was rushing to the school, late as usual. As I hustled up four flights of stairs to their classroom, another parent interrupted my thoughts and started talking. 

“She’s like me, basically,” the woman said. “She’s just not a math kid. We are creative types.”

I looked up, startled; I couldn’t hide my reaction. Here was a mom, already ruling out an entire world of possibilities for her child whose education had barely begun. Imagine if we treated reading in the same manner. 

The experience I had at pick-up is far from unique. As a math learning expert, I understand how deeply ingrained the myth of the math kid is in our education system. We classify or sort kids based on our perception of their varied, inborn math ability—"math kids” on one side, everyone else on the other.  

This view ignores the science that says all humans have an inherent number sense and ability to think mathematically from the start. In fact, scientists have proven that babies and toddlers show and develop numeracy—the ability to understand and work with numbers—early on. Babies only a few days old can distinguish two from three.

As school starts again, it’s time to bust the myth of the math kid. In particular, the widespread misconceptions that speed in math is paramount, math is a series of tricks, and only a single way exists to solve a problem. 

Myth # 1: Math is only about speed

If you’re like most Americans, you’ll remember the joy (the terror!) of timed multiplication tests. A sprint to the finish. Even if you did well on these timed tests, the moment the teacher said, “Begin,” hearts raced and stomachs churned. 

Let me be clear: The ability to call upon key facts and skills quickly— developing  fluency—is vital in math . It frees up working memory to tackle a hard problem in front of us. Early on, however, this type of overemphasis on speed ends up convincing students that math is only for the fastest students.

Additionally, as students progress to more advanced mathematics, they will need more than speed. They need the ability to approach problem solving in a calm, methodical way to ensure accuracy . 

Take for instance a study by Dartmouth College President Sian Leah Beilock, Ph.D., comparing two groups tasked with solving problems under time pressure: one group consisted of physics graduate students and professors, and the other, undergraduates who had completed just one physics class. Researchers assumed that the graduate students and professors would finish more quickly and accurately. To no one's surprise, the graduate students and professors were much more accurate. They, however, took longer to complete the problems. Their more rigorous approach involved a long, upfront pause to deeply understand the problem and consider the best approach before diving into problem solving. Read More: How to Help Your Kid With Math Even if You Suck at it

In overemphasizing speed, kids assume they don’t belong in math—reinforcing the myth of the math kid—and don’t advance their problem-solving abilities. Speed counts for something—just not for everything.

Myth # 2: Math is a series of tricks

Teaching math as a set of tricks tells kids that we don’t think they can understand what’s on the board. As with focusing too much on speed, this approach de-prioritizes any chance for deep understanding of math concepts and development of their problem-solving abilities. 

Take “anything times zero is zero.” This is a common trick we have kids memorize. Why is that true? Most kids will say, “It’s just a rule.” In fact, most adults will too. This unquestioning acceptance not only limits understanding, but also undermines common sense.

Consider instead visualizing this multiplication problem as cookies on plates. 3 × 1 means you have 3 plates, each plate with 1 cookie. That is 3 cookies altogether. And 3 × 3 means you still have 3 plates, but each plate has 3 cookies. That is 9 cookies altogether.

Now express 3 × 0 in plates and cookies. Okay, that is 3 plates. And how many cookies are on each plate? Zero. So now instead of 3 plates with 3 or 9 warm, gooey cookies, I have zero cookies in total. No cookies!

This visualization helps kids understand math more deeply, ensuring durable comprehension, which, unlike tricks, students can rely on.  Students can also learn practical skills like estimating answers or developing an intuition for when we are wrong. This approach demystifies math, making it both accessible and rigorous to all students, not just the so-called "math kids."

Myth # 3: There is only one way to do math

Finally, kids often believe that only one correct way exists to solve a problem and focus on getting to a quick answer. When they inevitably get stuck, this misconception discourages them from exploring alternative ways to solve a problem.

Consider a middle-school word problem: A store is selling 6 bags of marbles for $18. What is the unit price for a bag of marbles?   I have visited hundreds of math classrooms across the nation. Many times when a kid gets this type of question, they look up and ask: “Does ‘of’ mean multiplication?” This search for keywords like “of” is in the name of looking for a “single way” to solve the problem.

Real-world problem-solving involves exploring various strategies. For instance, if you lose your keys, you might retrace your steps, visualize where you last had them or check unusual places. Some of these may lead to dead ends, but a variety of techniques are likely to lead to finding your keys.

Let's reimagine the marble problem. Instead of looking for keywords, imagine a picture in your mind. A store has a bin full of marble bags and a sign that reads "6 bags for $18." If interested in just one bag, a customer would naturally calculate the price by dividing $18 by 6. Here, "of'' simply describes what's in the bags — it does not indicate multiplication. This kind of thinking shifts the focus to understanding, highlighting multiple ways to approach a problem.

When the preschool mom casually dismissed her daughter’s math ability, she may not have been thinking about these myths, but she was parroting the prevailing narrative of the math kid. But it doesn’t have to be this way. Instead, we can give kids—even preschoolers—the chance to hone their problem-solving skills, to develop deep understanding, and to utilize their inborn ability to think mathematically. Because, in reality, all kids are math kids.

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Humorous mathematical essays

Even though there are plenty examples of mathematical jokes , the mathematical literature is (in many cases) pretty dull. Nevertheless, examples exist in which an essay makes you smile with a nice pun when talking about Galois theory or something...

Can you provide examples of humorous/funny mathematical essays?

EXAMPLE ANSWER

A classical example is the essay "A Contribution to the Mathematical Theory of Big Game Hunting" by Ralph P. Boas (under a pseudonym) which was published in the Mathematical Monthly

Building on the comments by @Ahmed Hussein and @Hans Ludmark there is already a list of colorful language and a list of memorable titles so I changed my question.

• reference-request
• soft-question
• examples-counterexamples

• $\begingroup$ Looks like a horseshoe to me. $\endgroup$ –  copper.hat Commented Nov 26, 2015 at 7:03
• 5 $\begingroup$ There was once an article about self-referential structures in, IIRC, American Mathematical Monthly. Its list of references had a single item, the article itself. The only theorem of the article boldly stated that Theorem 1. in reference [1] is false. Where they usually place a photo of the author(s), there was a selfie taken with the aid of a mirror, the subject shown taking a picture et cetera... $\endgroup$ –  Jyrki Lahtonen Commented Nov 26, 2015 at 7:34
• 2 $\begingroup$ The margin notes in Concrete Mathematics , perhaps. Or the Romeo and Juliet interpretation of linear ODEs in Strogatz's Nonlinear Dynamics and Chaos (Section 5.3, especially the answers to exercise 5.3.1). $\endgroup$ –  Hans Lundmark Commented Nov 26, 2015 at 7:49
• 1 $\begingroup$ Check out: mathoverflow.net/questions/22299/… $\endgroup$ –  user258700 Commented Nov 26, 2015 at 7:50
• 1 $\begingroup$ Or some of the titles mentioned here: mathoverflow.net/questions/44326/most-memorable-titles . $\endgroup$ –  Hans Lundmark Commented Nov 26, 2015 at 7:50

Somewhere between a joke and an essay, is Impure Math. http://www.snowman-jim.org/science/humor/impure-math.html

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Analysis Based on factual reporting, although it incorporates the expertise of the author/producer and may offer interpretations and conclusions.

Making Equity Part of the Equation in Math Education

Math education outcomes in the United States have been unequal for decades. Learners in the top 10% socioeconomically tend to be about four grade levels ahead of learners in the bottom 10%—a statistic that has remained stubbornly persistent for 50 years.

To advance equity, policymakers and educators often focus on boosting test scores and grades and making advanced courses more widely available. Through this lens, equity means all students earn similar grades and progress to similar levels of math .

With more than three decades of experience as a researcher, math teacher, and teacher educator, I advocate for expanding what equity means in mathematics education. I believe policymakers and educators should focus less on test scores and grades and more on developing students’ confidence and ability to use math to make smart personal and professional decisions. This is mathematical power—and true equity.

What Is “Equity” in Math?

To understand the limitations of thinking about equity solely in terms of academic achievements, consider a student whom I interviewed during her freshman year of college.

Jasmine took Algebra 1 in ninth grade, followed by a summer online geometry course. This put her on a pathway to study calculus during her senior year in an AP class in which she earned an A. She graduated high school in the top 20% of her class and went to a highly selective liberal arts college. Now in her first year, she plans to study psychology.

Did Jasmine receive an equitable mathematics education? From an equity-as-achievement perspective, yes. But let’s take a closer look.

Jasmine experienced anxiety in her math classes during her junior and senior years in high school. Despite strong grades, she found herself “in a little bit of a panic” when faced with situations that require mathematical analysis. This included deciding the best loan options.

In college, Jasmine’s major required statistics. Her counselor and family encouraged her to take calculus over statistics in high school because calculus “looked better” for college applications. She wishes now she had studied statistics as a foundation for her major and for its usefulness outside of school. In her psychology classes, knowledge of statistics helps her better understand the landscape of disorders and to ask questions like, “How does gender impact this disorder?”

These outcomes suggest Jasmine did not receive an equitable mathematics education, because she did not develop mathematical power. Mathematical power is the know-how and confidence to use math to inform decisions and navigate the demands of daily life—whether personal, professional, or civic. An equitable education would help her develop the confidence to use mathematics to make decisions in her personal life and realize her professional goals. Jasmine deserved more from her mathematics education.

The Prevalence of Inequitable Math Education

Experiences like Jasmine’s are unfortunately common. According to one large-scale study, only 37% of U.S. adults have mathematical skills that are useful for making routine financial and medical decisions.

A National Council on Education and the Economy report found that coursework for nine common majors, including nursing, required relatively few of the mainstream math topics taught in most high schools. A recent study found that teachers and parents perceive math education as “ unengaging, outdated, and disconnected from the real world .”

Looking at student experiences, national survey results show that large proportions of students experience anxiety about math class , low levels of confidence in math, or both. Students from historically marginalized groups experience this anxiety at higher rates than their peers. This can frustrate their postsecondary pursuits and negatively affect their lives.

How to Make Math Education More Equitable

In 2023, I collaborated with other educators from Connecticut’s professional math education associations to author an equity position statement . The position statement, which was endorsed by the Connecticut State Board of Education, outlines three commitments to transform mathematics education.

1. Foster positive math identities .

The first commitment is to foster positive math identities, which includes students’ confidence levels and their beliefs about math and their ability to learn it. Many students have a very negative relationship with mathematics. This commitment is particularly important for students of color and those new to the English language to counteract the impact of stereotypes about who can be successful in mathematics.

A growing body of material exists to help teachers and schools promote positive math identities. For example, writing a math autobiography can help students see the role of math in their lives. They can also reflect on their identity as a “math person.” Teachers should also acknowledge students’ strengths and encourage them to share their own ideas as a way to empower them.

2. Modernize math content .

The second commitment is to modernize the mathematical content that school districts offer to students. For example, a high school mathematics pathway for students interested in health care professions might include algebra, math for medical professionals, and advanced statistics. With these skills, students will be better prepared to calculate drug dosages, communicate results and risk factors to patients, interpret reports and research, and catch potentially life-threatening errors.

3. Align state policies and requirements .

The third commitment is to align state policies and school districts in their definition of mathematical proficiency and the requirements for achieving it. In 2018, for instance, eight states had a high school math graduation requirement insufficient for admission to the public universities in the same state. Other states’ requirements exceed the admission requirements. Aligning state and district definitions of math proficiency clears up confusion for students and eliminates unnecessary barriers.

What’s Next?

As long as educators and policymakers focus solely on equalizing test scores and enrollment in advanced courses, I believe true equity will remain elusive. Mathematical power—the ability and confidence to use math to make smart personal and professional decisions—needs to be the goal.

This article is republished from The Conversation under a Creative Commons license. Read the original article .

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The Benefits of Everyday Math for Kids

• Mathematical Ability

APS’s Özge Gürcanlı Fischer Baum chats with Melissa Libertus from University of Pittsburgh about her new article about interventions to increase math learning in children. They discuss various strategies parents can use to reinforce the development of math skills in everyday life like at the grocery store or using board games. 

Send us your thoughts and questions at  [email protected] .

Unedited transcript

[00:00:00.000] – APS’s Özge Gürcanlı Fischer Baum

Having numerical skills is important in life. Before children start school, there is significant variability in their mathematical skills. What can be done at home to support children? What type of parental activities can make a difference in their math proficiency? What are the learning opportunities out there? This is under Under the Cortex, I am Özge Gürcanlı Fischer Baum, with the Association for Psychological Science. To speak about activities designed to facilitate math learning at home for young children, I have with me Melissa Libertus from University of Pittsburgh. She is the author of an article published in Current Directions in Psychological Science, examining the Relationship between foundational numeracy skills and parental Interventions. Melissa, thank you for joining me today. Welcome to Under the Cortex.

[00:01:00.520] – Melissa Libertus

Thank you. Great to be here.

[00:01:02.900] – APS’s Özge Gürcanlı Fischer Baum

Yeah, we are very happy to have you, too. I want to start asking, what got you interested in studying children and how they learn?

[00:01:11.830] – Melissa Libertus

This is a really good question, and it started I did it a long time ago. I first started tutoring when I was in middle school. At first, it was just other students in my class who came over to my house after school to get help with homework. Over time, the teachers asked me to help students in other classes and who were struggling. And by the time I finished high school, I probably had about 10 to 12 students who regularly saw me for extra lessons, and most of them needed help in math. And what fascinated me about this was that for some students, It was just some extra practice that they needed. And other students, they really struggled understanding concepts. And so I came up with ways to explain the same concepts in different ways to them until it finally clicked. I got really interested in how it is that children sometimes struggle understanding different concepts, especially in math, and how we can best support them to learn what they need to succeed in the long run. I really wanted to know more about what these foundations are that set up children for success in math.

[00:02:26.960] – APS’s Özge Gürcanlı Fischer Baum

Right. First, I want to say it’s an impressive number of students that you held when you were younger, and it feels like it was like a natural experimental setting for you to observe differences in your friends at the time. Yeah, thank you for sharing that story with us. I want to continue with a very generic question, but I think it is important for us to have these conversations. Why is early math literacy for children important? What is the early stage, according to developmental researchers?

[00:03:02.040] – Melissa Libertus

Yes. Ample research has shown that early math skills are predictive of a host of later life outcomes. For example, Richie and Bates showed that early math abilities at age seven predict socioeconomic status at age 42, even when you control for socioeconomic status at birth. And similarly, Pamela Davis-Keen and colleagues showed that math skills at four and a half years of age predict later high school math class and ultimately college enrollment. And importantly, math skills in this study were measured before children start formal schooling, which emphasizes the importance of fostering math skills long before children start learning in a formal setting. So what do these math skills prior to elementary school look like? Math concepts for very young children start from birth. So research by Fais Hsu and Laspalki and others in the early 2000s showed that even young infants possess a basic understanding of quantities already. For example, if you repeatedly show a young infant images that have the same number of objects and they get habituated to seeing, say, images that have eight dots on them over and over, and then you show them a picture that contains 16 dots, they tend to look longer at those.

[00:04:27.540] – Melissa Libertus

What we know is that these A basic understanding of quantities is something that children have long before they acquire the verbal skills and the language to really describe what these quantities are and to count them, obviously. And so 10 years ago, Ariel Star, Liz Bryant and I, showed that individual differences in infants’ early approximate number abilities is predictive of children’s math abilities at the age of three and a half. And then as children acquire language, they slowly learn the meaning of a number of words. And most of the documented evidence for this development currently comes from toddlers who learn English. And for example, in her seminal work, Karen Wyn showed that around 30 months of age, many children are able to correctly give one object when you ask them for exactly one. But they are unable to give two or three objects when you’re asking them for two or three, and instead, they might just grab a handful. And it takes a couple of months after learning the meaning of this number one, that children are then actually able to also correctly give two objects when they asked for two, and then it takes several more months until they can correctly give three objects when they asked for three.

[00:05:48.040] – Melissa Libertus

It takes a really long time for children to get this exact understanding of these number words. It’s around four years of age that many English-speaking children are then finally able to create sets of specific numbers of objects when given a specific number word. Those are the early stages of math learning, but there are other things to it, too. Young kids do learn to understand what more and less means, and they learn to see patterns in shapes and numbers around them.

[00:06:26.480] – APS’s Özge Gürcanlı Fischer Baum

You gave us a list of very important pieces of information. Let me repeat them for our audience. We are clearly born with some skills for quantitative knowledge. But from research, we know that the challenge lies in tying it to our language. When children do nonverbal tasks, they are quite skilled at these numerical skills. But when we see them using language with maths, it is a slower process and slow progress, in fact, for them to get to regular counting or using numbers in the ways that we do. What I hear is that everything is interconnected. Even though we come to this world with some quantitative skills, there are challenges ahead of us, which brings us back to your interventions, I guess. Let’s I’ll talk about them a little bit. So your paper is very comprehensive. I really enjoyed reading it. You go over two different theories that scientists use. Can you go over what they are?

[00:07:44.670] – Melissa Libertus

So there are two main theories that play a role in thinking about math interventions, especially in the context of early childhood and the home setting. The first one is the Opportunity Propensity Model, and the second one is the Sociocultural Theory of Development. The Opportunity Propensity Model was originally put forth by James Burns and colleagues, and it suggests that academic achievement is a product of opportunities to acquire content knowledge and practice this knowledge, as well as a learner’s propensity to benefit from these opportunities. In the context of math learning, this means that children need opportunities to learn math concepts. We might be born with a very basic, rudimentary understanding of quantities, but we really need opportunities to sharpen and really arrive at an exact understanding of numbers. This can happen in a classroom where a teacher introduces the concept, But also it can happen in the home when a parent guides a child. For example, you might be counting how many blocks they each have and who has more and who has less. It can happen during everyday situations, like when a child finds a penny on a street and the parent points out that this is a penny and it’s worth less than a dime, for example.

[00:09:08.360] – Melissa Libertus

But at the same time, this opportunity propensity model also highlights that children have to bring some propensity some skills to the table. They need domain general cognitive skills. Obviously, they need to have language skills. They need to be attentive to remember what was said. So attention and working memory are key. But they also need to have some basic understanding of numbers than to build concepts like more and less. Then on the other hand, we have the sociocultural theory of development championed by Barbara Róhoff and others that It’s that children learn through participation in everyday activities that are embedded in their communities. Children grow up in communities where people use math for different purposes. Children might participate in meal preparation in their family, which requires to set the table and count if there are enough plates and utensils for everyone. They might help with cooking and measuring ingredients. They might go shopping or maybe even sell family’s goods and thereby learn about the value of currency. Children’s opportunities to learn math are tightly coupled to the cultural context in which they grew up and how math may be embedded into that setting.

[00:10:26.350] – APS’s Özge Gürcanlı Fischer Baum

When I think about these cultural opportunities, in a way, I’m always thinking about Halloween or other holidays in Turkey during Eid. I used to receive candy, and my brother, he’s older, tried to take them away from me, and I had to be really good at math, knowing how many candy pieces I had. So yeah, great theories, and you explained them really, very nicely for us. In your paper, you talk about different types of interventions that parents could use at home to help their children to think more about math, to use math more? Which would you say is your favorite and why?

[00:11:14.250] – Melissa Libertus

My favorite and why? My personal favorite is to integrate math into grocery shopping. I think there are so many opportunities, and they can also easily scale with the child’s math abilities. Initially, with a young child, grocery grocery shopping provides an opportunity to count the number of tomatoes you want to buy or the number of apples in a bag, or to identify Arabic numerals on prices and to point them out. One can use this also as an opportunity to compare weights of items and to weigh the bag of apples to see, is this really three pounds in the bag? Or to take different-size tomatoes and see how much each Then as children get older, shopping provides a great opportunity to learn about money, which is a really complicated concept for kids initially, because why is it that one green bill is only worth $1 because there’s a one printed on it, but then I can have another bill that has a two and a zero printed on it, and that’s now $20. It’s also just one piece of paper, so to speak. Teaching children these abstract values and obviously being able to do arithmetic in the context of money.

[00:12:38.590] – Melissa Libertus

Then as children become even older, there are decimals in prices. You can figure out how much something What does something cost if there is a 10% discount on it. I think there are just a whole host of opportunities for children to practice math skills in this context and to really also see how important it to know math and to be able to do it exactly. It’s not helpful if you can do it, but you actually should know if the change that you got back is correct or not.

[00:13:13.080] – APS’s Özge Gürcanlı Fischer Baum

If you want food, you need to know your numbers, basically. Yeah, that’s a great example. There are other examples, options of interventions in your paper. One option was a numbers-based board game. What games are those and How do they learn and practice numbers using this particular game?

[00:13:35.220] – Melissa Libertus

Yes, most of the board games that have been used in research studies are actually very similar to shoots and letters and other commercially available board games like this, where you have a player roll a die or spin a spinner and advance across the board, and whoever reaches the goal first wins. But what’s key here is that along the way, children can practice counting number of steps they are allowed to advance. In one of our own studies, my postdocs, Andy Ripner and Leon Elliott, as well as my graduate student, Alex Silver, asked parents to play a number board game twice a week for eight weeks. On this board game that we used, we had 65 fields, and parents were instructed to label the numbers that were on these spots as they and their children were moving their tokens across the board. If you had your token on the spot number 23, for example, and you spun a three, you would say 24, 25, 26. Children not only counted and learned the small numbers that were on the spinner, but children also read and heard the labels of these double-digit numbers that were printed on the board.

[00:14:47.990] – Melissa Libertus

We think that those are natural ways where kids can practice age-appropriate counting and numeral literacy skills, where they just get to repeatedly read and count and say out these number words in the context of playing a fun game with a parent.

[00:15:07.310] – APS’s Özge Gürcanlı Fischer Baum

You said they did this intervention twice a week for a long time, and the outcome was helping children, right?

[00:15:18.510] – Melissa Libertus

Yes, exactly. In the study, we found that children who played this board game with their parents repeatedly for these eight weeks had greater math skills after this intervention compared to children who played a board game that didn’t include this number P. They had different shapes on the board instead, and they spun a spinner that told them which shape to advance to next. A very similar game, yet it did not have the number-based focus.

[00:15:53.660] – APS’s Özge Gürcanlı Fischer Baum

Yeah, that’s impressive because two months is not a long time. The The game sounds like fun and it is quality family time. They get to hear numbers and it helps with their math skills. That is what is not to like about this intervention. Yes. There’s another one. Another strategy was reading number-based books, but for this one, benefits didn’t last at the two-month follow-up, right? Why do you think that is? Do you think the children didn’t retain knowledge or that the control group caught up? Or why do you think this is different from the board game that we just talked about?

[00:16:37.870] – Melissa Libertus

The specific study that you were referring to here was done by David Purpura and his group at Purdue University. They asked parents to read picture books with their children four times for four weeks. Half of them received books with math content, and the other half read similar books without the math content. What they found is that children in the a math book group showed significant improvements in comprehension of math language, so words like more and less, least, and enough, immediately after the intervention compared to those children who read the other books that didn’t have math content. However, then this effect was no longer present eight weeks later. But when they did detailed follow-up analysis, they found that there were still improvements, even two months later, when they looked at the specific math language that was covered in the picture books. And so there are some benefits that do get retained. Importantly, I should point out that children in the math book group also showed significant improvements on a standardized math assessment compared to the control group, and that effect was present immediately after the intervention and also still eight weeks later. So I think in some of these findings suggest that children do indeed benefit from reading such math picture books and that these effects can last beyond the immediate intervention period.

[00:18:12.000] – APS’s Özge Gürcanlı Fischer Baum

The benefits can vary, right? Depending on what the activity is. There are multiple ways to test the benefit after these interventions. Exactly. Yeah, thank you. Thank you for your explanation for that. Let’s talk about parents’ attitudes towards math a little bit because not everybody loves math. This is, I think, something true across cultures. How do parents’ attitudes towards math impact their children’s math learning?

[00:18:42.660] – Melissa Libertus

Yes, this is an excellent and somewhat understudied topic because there are a number of different aspects that need to be considered, too. For example, as you rightly point out, parents’ own affect toward math plays a role. Some people experience math anxiety, for example. They get really nervous or show negative physical responses when they have to solve math problems. And there’s been some really interesting research by Susan Labine and colleagues at the University of Chicago that show that parents with greater math anxiety tend to have children who perform worse in math. And importantly, they found that parents’ math anxiety is particularly detrimental if math-anxious parents help their children with math homework. But on the other side, they also found found that when parents who were math-anxious were given explicit guidance on how to engage with their children by using a math story app, that those children’s math performance then increased as much as the math performance of children of low math-anxious parents. Parents’ attitudes can impact children’s math learning, but that can be offset by giving parents the right guidance and tools to overcome that anxiety, at least at these young ages that we’re talking about today.

[00:20:06.940] – Melissa Libertus

But then I think math anxiety is also just one part of the story, because I think the other part is really that parents’ beliefs about the importance of math and other related beliefs. How about who’s responsible to teach a child math or what role the school plays versus the home? That’s important, too. For example, my graduate student, Alex Silver, and my postdoc, Leon Elliott, and I found that parents’ beliefs about the importance of math interacts with their math anxiety. We found that parents with high math anxiety who believed that math was particularly important, had preschoolers with above average math performance, whereas parents who rated math as less important had children with lower than average math performance. And in contrast, if you looked at parents with low math anxiety, the beliefs were not associated with children’s math outcome. So what I think is key here is that when parents have really strong beliefs about the importance of math, then they are trying to seek out opportunities for their child to learn math, even if they themselves don’t feel comfortable, or especially if they themselves have math anxiety because they might not want to transmit that anxiety to their children.

[00:21:28.290] – APS’s Özge Gürcanlı Fischer Baum

What I hear from I guess the first part of your answer is that parents themselves sometimes need an intervention about their anxiety about this. If it is addressed well, it benefits both the parents and also children. It is something important to know for our listeners.

[00:21:48.030] – Melissa Libertus

[00:21:49.030] – aps’s özge gürcanlı fischer baum.

I would like to ask you a little bit more about what science says in general about these interventions. Is there a consensus about what How does early interventions work? Should we be trying some different approaches? What are your thoughts about that?

[00:22:07.630] – Melissa Libertus

I think that research demonstrates that parents are indeed capable of providing children with opportunities to learn math, and especially increase that when given appropriate instructions or materials, and that those interventions often lead to improvements in children’s math. I think it’s really important to highlight that parents don’t need a lot of training or extra time or resources to provide their children with opportunities to learn math. I think that’s the beauty that math is everywhere and can easily be integrated into all sorts of different activities. My Bachelor of Philosophy student, Erin Hannah, for example, showed that simply putting up signs in grocery stores that provided adults with simple prompts for math conversations with young kids, such as saying, How many eggs are in a carton? Or, How How many would be left if we each ate two? Are sufficient to get more adults to talk about math with their children while they are shopping. And importantly, Sue Hasposs and I replicated these findings in a food pantry, showing that this can be done anywhere and everywhere. So I think this is really key that we just need to nudge adults and parents just a little bit to see the opportunities to integrate math into the everyday activities that they are already doing with children.

[00:23:30.960] – Melissa Libertus

I think that failures to sometimes find improvements in children’s math abilities in some of those rigorous research studies are no reason to abandon these interventions or these ideas. Instead, I think that future research should carefully consider how children’s skills align with the intervention activities and which families in which contexts may benefit the most from the interventions. I think making the interventions meaningful for families’ lives and something that they see value in, that they can integrate into things that they are already doing, are key to really giving kids the opportunities and in the long run, benefiting from it. I think The other piece that’s tricky with some of these intervention studies is that oftentimes, durations and dosages may not be aligned with what’s best to find the effects. Then obviously, it also takes time to consolidate the skills and for children to integrate the skills that they may have learned through these interventions. Different children may take different amounts of time to do so. Oftentimes, these intervention studies have a one-size-fits-all. You’re doing this four times a week for four weeks. You’re doing this for eight weeks. That might not work for everybody.

[00:24:53.560] – APS’s Özge Gürcanlı Fischer Baum

I want to just repeat what you said because it is very important. There is no one-size-fits-all solution solution to this. We should pay attention to what families, in what context we should address. All these different interventions and approaches you have been talking about, focus on making this part of everyday conversation. It doesn’t have to be a separate math time that needs to happen in already busy schedule lives of parents.

[00:25:26.730] – Melissa Libertus

Exactly. Absolutely, yes.

[00:25:29.250] – APS’s Özge Gürcanlı Fischer Baum

Yeah. I would like to ask if you have anything else to add to share with our listeners about these parental interventions.

[00:25:41.160] – Melissa Libertus

Yes, absolutely. I think if you have young children yourself or regularly interact with them, find ways to point out math, see the opportunities to make comparisons like more and less of finding patterns, identifying shapes, counting forwards and backwards, we’re skip counting, we’re making up arithmetic problems. I think we also need to raise people’s awareness that it’s important to integrate math concepts in these conversations, that it’s not too early to start exposing young children to math, that we don’t have to wait for them to start kindergarten or first grade. Because these days, more than ever, we need people with problem solving skills and STEM skills, and those are the people who ultimately get the jobs that are out there. We need to prepare our children today to be ready to join the workforce tomorrow.

[00:26:43.210] – APS’s Özge Gürcanlı Fischer Baum

Yeah, let’s make Matt part of everyday conversation to support children and to support their quantitative skills. Absolutely. Melissa, thank you very much. This was a pleasure. All these questions you answered, I would like to thank on behalf of our listeners.

[00:27:03.960] – Melissa Libertus

My pleasure. Thank you so much.

[00:27:06.640] – APS’s Özge Gürcanlı Fischer Baum

This is Özge Gürcanlı Fischer Baum with APS, and I have been speaking to Melissa Libertus from University of Pittsburgh. If you want to know more about this research, visit psychologicalscience.org. Would you like to reach us? Send us your thoughts and questions at [email protected] .

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