math problem solving essay

How mathematical practices can improve your writing

Writing is similar to three specific mathematical practices: modelling, problem-solving and proving, writes Caroline Yoon. Here, she gives some tips on how to use these to improve academic writing

Caroline Yoon

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I feel for my students when I hand them their first essay assignment. Many are mathematicians, students and teachers who chose to study mathematics partly to avoid writing. But in my mathematics education courses, and in the discipline more generally, academic writing is part of our routine practice.

Mathematicians face some challenging stereotypes when it comes to writing. Writing is seen as ephemeral, subjective and context-dependent, whereas mathematics is seen as enduring, universal and context-free. Writing reflects self, but mathematics transcends it: they are distinct from each other.

This is a false dichotomy that can discourage mathematicians from writing. It suggests writing is outside the natural skill set of the mathematician, and that one’s mathematics training not only neglects one’s development as a writer but actively prevents it.  Rather than capitulate to this false dichotomy, I propose we turn it around to examine how writing is similar to three specific mathematical practices: modelling, problem-solving and proving.

Three mathematical practices that can improve your writing

Mathematical modelling.

Let us consider a hypothetical mathematics education student who has spent weeks thinking, reading and talking about her essay topic, but only starts writing it the night before it is due. She writes one draft only – the one she hands in – and is disappointed with the low grade her essay receives.

She wishes she had started earlier but she was still trying to figure out what she wanted to say up until the moment she started writing. It was only the pressure of the deadline that forced her to start; without it, she would have spent even more time thinking and reading to develop her ideas. After all, she reasons, there is no point writing when you do not know what to write about!

This “think first, write after” approach, sometimes known as the “writing up” model is a dangerous trap many students fall into, and is at odds with the way writing works. The approach allows no room for imperfect drafts that are a necessary part of the writing process. Writing experts trade on the generative power of imperfect writing; they encourage writers to turn off their internal critics and allow themselves to write badly as a way of overcoming writing inertia and discovering new ideas. The “shitty first draft” is an ideal (and achievable) first goal in the writing process. Anyone can produce a sketchy first draft that generates material that can be worked on, improved and eventually rewritten into a more sharable form.

Mathematical modelling offers a compelling metaphor for the generative power of imperfect writing. Like polished writing, polished mathematical models are seldom produced in the first attempt. A modeller typically begins with some understanding of the real situation to be modelled. The modeller considers variables and relationships from his or her understanding of the real situation and writes them into an initial mathematical model.

The model is his or her mathematical description of the situation, written in mathematical notation, and the modeller who publishes a mathematical model has typically created and discarded multiple drafts along the way, just as the writer who publishes a piece of writing has typically written and discarded multiple drafts along the way.

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Problem-solving

Writing an original essay is like trying to solve a mathematics problem. There is no script to follow; it must be created by simultaneously determining one’s goals and figuring out how to achieve them. In both essay writing and mathematical problem-solving, getting stuck is natural and expected. It is even a special kind of thrill.

This observation might come as a surprise to mathematicians who do not think of their problem-solving activity as writing. But doing mathematics, the ordinary everyday act of manipulating mathematical relationships and objects to notice new levels of structure and pattern, involves scratching out symbols and marks, and moving ideas around the page or board.

Why do I care that mathematicians acknowledge their natural language of symbols and signs as writing? Quite frankly because they are good at it. They have spent years honing their ability to use writing to restructure their thoughts, to dissect their ideas, identify new arguments. They possess an analytic discipline that most writers struggle with.

Yet few of my mathematics education students take advantage of this in their academic writing. They want their writing to come out in consecutive, polished sentences and become discouraged when it does not. They do not use their writing to analyse and probe their arguments as they do when they are stuck on mathematical problems. By viewing writing only as a medium for communicating perfectly formed thoughts, they deny themselves their own laboratories, their own thinking tools.

I am not suggesting that one’s success in solving mathematical problems automatically translates into successful essay writing. But the metaphor of writing as problem-solving might encourage a mathematics education student not to give up too easily when she finds herself stuck in her writing.

Our hypothetical student now has a good draft that she is happy with. She is satisfied it represents her knowledge of the subject matter and has read extensively to check the accuracy of its content. A friend reads the draft and remarks that it is difficult to understand. Our student is unperturbed. She puts it down to her friend’s limited knowledge of the subject and is confident her more knowledgeable teacher will understand her essay.

But the essay is not an inert record judged on the number of correct facts it contains. It is also a rhetorical act that seeks to engage the public. It addresses an audience, it tries to persuade, to inspire some response or action.

Mathematical proofs are like expository essays in this regard; they must convince an audience. When undergraduate mathematics students learn to construct proofs of their own, a common piece of advice is to test them on different audiences. The phrase “Convince yourself, convince a friend, convince an enemy” becomes relevant in this respect.

Mathematicians do not have to see themselves as starting from nothing when they engage in academic writing. Rather, they can use mathematical principles they have already honed in their training, but which they might not have formerly recognised as tools for improving their academic writing.

Practical tips for productive writing beliefs and behaviours

• Writing can generate ideas. Free writing is a good way to start. Set a timer and write continuously for 10 minutes without editing. These early drafts will be clumsy, but there will also be some gold that can be mined and developed.
• Writing can be used to analyse and organise ideas. When stuck, try to restructure your ideas. Identify the main point in each paragraph and play around with organising their flow. 
• Writing is a dialogue with the public. Seek out readers’ interpretations of your writing and listen to their impressions. Read your writing out loud to yourself: you will hear it differently!

Caroline Yoon is an associate professor of mathematics at the University of Auckland.

This is an edited version of the journal article “The writing mathematician” by Caroline Yoon, published in For the Learning of Mathematics  and collected in The Best Writing on Mathematics , edited by  Mircea Pitici  (Princeton University Press).

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Composition Type: Problem-Solution Essays

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• Ph.D., Rhetoric and English, University of Georgia
• M.A., Modern English and American Literature, University of Leicester
• B.A., English, State University of New York

In composition , using a problem-solution format is a method for analyzing and writing about a topic by identifying a problem and proposing one or more solutions. A problem-solution essay is a type of argument. "This sort of essay involves argumentation in that the writer seeks to convince the reader to take a particular course of action. In explaining the problem, it may also need to persuade the reader concerning specific causes" (Dave Kemper et al., "Fusion: Integrated Reading and Writing," 2016).

The Thesis Statement

In many types of report writing, the thesis statement is posed front and center, in one sentence. Author Derek Soles writes about how the thesis statement in a problem-solution paper differs from a straight "report of findings" type of text:

"[One]  expository  mode is the problem-solution essay, topics for which are typically framed in the form of questions. Why did fourth-graders from poor families score low on a nationwide math test, and how can educators improve math education for this group? Why is Iran a threat to our national security, and how can we reduce this threat? Why did it take the Democratic Party so long to select a candidate for the 2008 presidential election, and what can the party do to make the process more efficient in the future? These essays have two parts: a full explanation of the nature of the problem, followed by an analysis of solutions and their likelihood of success."

("The Essentials of Academic Writing," 2nd ed. Wadsworth, Cengage, 2010)

Readers need additional context before you get to your thesis, but that is not to say that the thesis has to be posed as a question in the introduction:  

"In a problem-solution essay, the thesis statement usually proposes the solution. Because readers must first understand the problem, the thesis statement usually comes after a description of the problem. The thesis statement does not have to give details about the solution. Instead, it summarizes the solution. It should also lead naturally to the body of the essay, preparing your reader for a discussion of how your solution would work."

(Dorothy Zemach and Lynn Stafford-Yilmaz, "Writers at Work: The Essay." Cambridge University Press, 2008)

Sample Introductions

It can be helpful to see completed examples before writing in order to examine what makes for an effective piece. See how these introductions give some context before posing the topic and lead naturally into the body paragraphs, where the evidence will be listed. You can imagine how the author has organized the rest of the piece.

"We buried my cousin last summer. He was 32 when he hanged himself from a closet coat rack in the throes of alcoholism, the fourth of my blood relatives to die prematurely from this deadly disease. If America issued drinking licenses, those four men—including my father, who died at 54 of liver failure—might be alive today."

(Mike Brake, "Needed: A License to Drink."  Newsweek , March 13, 1994)

"America is suffering from overwork. Too many of us are too busy, trying to squeeze more into each day while having less to show for it. Although our growing time crunch is often portrayed as a personal dilemma, it is, in fact, a major social problem that has reached crisis proportions over the past twenty years."

(Barbara Brandt, "Whole Life Economics: Revaluing Daily Life." New Society, 1995)

"The modern-day apartment dweller is faced with a most annoying problem: paper-thin walls and sound-amplifying ceilings. To live with this problem is to live with the invasion of privacy. There is nothing more distracting than to hear your neighbors' every function. Although the source of the noise cannot be eliminated, the problem can be solved."

(Maria B. Dunn, "One Man's Ceiling Is Another Man's Floor: The Problem of Noise")

Organization

In "Passages: A Writer's Guide, " how to organize a problem-solution paper is explained:  

"Though to some extent [your organization of the paper] depends on your topic, do make sure that you include the following information:

Introduction: Identify the problem in a nutshell. Explain why this is a problem, and mention who should be concerned about it.

Problem Paragraph(s): Explain the problem clearly and specifically. Demonstrate that this is not just a personal complaint, but a genuine problem that affects many people.

"Solution Paragraph(s): Offer a concrete solution to the problem, and explain why this is the best one available. You may want to point out why other possible solutions are inferior to yours. If your solution calls for a series of steps or actions to be followed, present these steps in a logical order.

"Conclusion: Reemphasize the importance of the problem and the value of your solution. Choose a problem that you have experienced and thought about—one that you have solved or are in the process of solving. Then, in the essay itself, you may use your own experience to illustrate the problem. However, don't focus all the attention on yourself and on your troubles. Instead, direct the essay at others who are experiencing a similar problem. In other words, don't write an I essay ('How I Cure the Blues'); write a you essay ('How You Can Cure the Blues')."

(Richard Nordquist, Passages: A Writer's Guide , 3rd ed. St. Martin's Press, 1995)

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Aspects of the Mathematics Problem Solving Essay

Introduction.

It is important to note that Jo Boaler is a writer and educator who has received many prizes and awards. At the same time, the woman leads a movement that aims to change the way teachers and students think about mathematics and perceive it. Thus, during her speech at TEDx, she talked about the experiment she conducted at Stanford. Moreover, the speaker immediately announced that her students cried in order to interest people to listen and capture their attention. Hence, the experiment was performed with students who spent a lot of time studying algebra and really tried to improve their knowledge. However, the innovation was that Jo Boaler did not use standard tests to check their knowledge, but showed them a virtual picture with squares and assigned tasks.

Interestingly, the students’ opinions were different, they believed that the squares grow horizontally, vertically, or proportionally. However, the speaker notes that two people in the group cried because they realized that the simple principle that explains the algebraic progression. Moreover, the tears were caused by the fact that these people spent many hours studying higher mathematics using standard methods, but most of them could not explain the arrangement of squares in the picture.

Then, the woman began to share her experience of studying at school, where children constantly learn new symbols and formulas, which makes them tired and afraid. Therefore, most of the students do not have enough knowledge of mathematics, because they consider this subject extremely difficult and boring. Instead, as a solution to this problem, Jo Boaler proposes to apply knowledge about the brain to make it easier to learn mathematics. Therefore, the speaker demonstrates a picture of the human brain and shows that two of the five brain pathways are visual. Thus, she showed that the brain can more easily perceive information that is difficult for it, which is presented with the help of pictures. In order to explain her argument, she gives an example of a rectangle and suggests using bright pictures to find all the rectangles depicted on them. The results show that this way is easier for perception and students offer many possible solutions. Thus, the use of visual mathematics is available to explain various topics, and resources for teacher or parent training can be found on the website of the university where Jo Boaler teaches.

In addition, the expert cites statistics as an argument for the effectiveness of virtual mathematics. Accordingly, a section with an attachment for studying visual mathematics appeared on the university’s website two years ago, and now almost three million teachers per month view these materials. This is because, in practice, teachers have become convinced that visual mathematics is a way to explain educational material to students in an accessible manner and to arouse their constant interest. Accordingly, this method allows to unlock the potential of even those students who believe that they do not have mathematical abilities. This is due to the fact that with the help of visual learning of mathematics, it is easier to solve complex problems and correct mistakes made, which provokes an increase in self-belief and self-confidence.

Thus, Jo Boaler explains why math can be such a difficult subject for some people, while other people can easily complete tasks. Moreover, she also shares the latest scientific research on the brain to demonstrate how mathematics is processed by the human brain. At the same time, the speaker demonstrated the importance of visual learning and the need for such a concept as self-confidence to obtain a positive experience of learning mathematics.

A Technique to Eliminate Math Anxiety

It is significant to mention that Dr. Katie Nall, who holds a PhD in philosophy, addressed the TEDx platform. At the same time, the speaker also has a mathematical education and received a doctorate in mathematics from the Florida Institute of Technology while working full-time. Therefore, the woman’s advice is extremely practical, because Dr. Katie Nall’s scientific research was based on the problems she faced in practice. Accordingly, the speaker, while writing her dissertation, studied the behavior of students who felt fear and anxiety in mathematics classes.

During the speech, the speaker with the use of humor notes that since childhood she loved and respected mathematics, but her students, even those who worked hard to pass the subject, hate this science. Hence, the speaker tried to give them mathematical advice, but this method did not bring the desired results. After that, the woman decided to explore a technique that helps to escape from anxiety caused by any other factors. The woman chose the technique of emotional freedom, which is a psychological method that enables people to get rid of negativity and unnecessary frameworks. Thus, they help to achieve the stated goal. At the same time, emotional freedom techniques are easy to use and require little time, although it is strictly individual. Dr. Katie Nall suggested her students apply this method in order to reduce fear and anxiety caused by mathematics.

The technique was to admit their mistakes and try to accept themselves with them. For this purpose, it is necessary to repeat the problematic moments, and in the end to note that you are safe. At the same time, it is crucial to tap on the points, moving along the same line, for example, the inner edge of the eyebrow, the outer corner of the eye, under the eye, under the nose, at the top of the chin, the junction of the collarbones, the top of the head. This is because there are harmonizing, calming, exciting, and signal points on the line, which all need to be activated. In addition, students who previously could not pass the subject, but now used this technique were able to overcome fear and received a satisfactory grade. Furthermore, Dr. Katie Nall notes that the students continued to use this tactic during their further studies, as they received successful outcomes.

Meanwhile, the speaker recalls cases when students thanked her for suggesting they try this technique. Some students hugged her, while others said that they also used her method outside the classroom. During the speech, the speaker also noted that this practice is common in many educational institutions, for example, in California, in some schools students and teachers allocate six waves a day to reduce stress and increase the ability of students to successfully complete school tasks. Moreover, the use of this technique is beneficial for students’ health. Since a large number of visits to doctors are associated with stress factors.

Hence, on the TEDx platform, Dr. Katie Nall offered a solution to eliminate math anxiety. At the same time, the speaker explained the importance of basic methods of dealing with other fears and worries. The emotional freedom technique is a powerful method of dealing with emotional problems that students often face when studying mathematics. Therefore, using the advice of a female is extremely valuable for people studying mathematics.

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NEW TJHSST MATH PROBLEM-SOLVING ESSAY WORKBOOK VOLUME 1: Mastering New TJHSST Math Problem-Solving Essay Paperback – January 25, 2021

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Essays About Math: Top 10 Examples and Writing Prompts 

Love it or hate it, an understanding of math is said to be crucial to success. So, if you are writing essays about math, read our top essay examples.  

Mathematics is the study of numbers, shapes, and space using reason and usually a special system of symbols and rules for organizing them . It can be used for a variety of purposes, from calculating a business’s profit to estimating the mass of a black hole. However, it can be considered “controversial” to an extent.

Most students adore math or regard it as their least favorite. No other core subject has the same infamy as math for generating passionate reactions both for and against it. It has applications in every field, whether basic operations or complex calculus problems. Knowing the basics of math is necessary to do any work properly. 

If you are writing essays about Math, we have compiled some essay examples for you to get started. 

1. Mathematics: Problem Solving and Ideal Math Classroom by Darlene Gregory 

2. math essay by prasanna, 3. short essay on the importance of mathematics by jay prakash.

• 4.  Math Anxiety by Elias Wong

5. Why Math Isn’t as Useless as We Think by Murtaza Ali

1. mathematics – do you love or hate it, 2. why do many people despise math, 3. how does math prepare you for the future, 4. is mathematics an essential skill, 5. mathematics in the modern world.

“The trait of the teacher that is being strict is we know that will really help the students to change. But it will give a stress and pressure to students and that is one of the causes why students begin to dislike math. As a student I want a teacher that is not so much strict and giving considerations to his students. A teacher that is not giving loads of things to do and must know how to understand the reasons of his students.”

Gregory discusses the reasons for most students’ hatred of math and how teachers handle the subject in class. She says that math teachers do not explain the topics well, give too much work, and demand nothing less than perfection. To her, the ideal math class would involve teachers being more considerate and giving less work. 

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“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.”

In her essay, Prasanna gives readers a basic idea of what math is and its importance. She additionally lists down some of the many uses of mathematics in different career paths, namely managing finances, cooking, home modeling and construction, and traveling. Math may seem “useless” and “annoying” to many, but the essay gives readers a clear message: we need math to succeed. 

“In this modern age of Science and Technology, emphasis is given on Science such as Physics, Chemistry, Biology, Medicine and Engineering. Mathematics, which is a Science by any criterion, also is an efficient and necessary tool being employed by all these Sciences. As a matter of fact, all these Sciences progress only with the aid of Mathematics. So it is aptly remarked, ‘Mathematics is a Science of all Sciences and art of all arts.’”

As its title suggests, Prakash’s essay briefly explains why math is vital to human nature. As the world continues to advance and modernize, society emphasizes sciences such as medicine, chemistry, and physics. All sciences employ math; it cannot be studied without math. It also helps us better our reasoning skills and maximizes the human mind. It is not only necessary but beneficial to our everyday lives. 

4.   Math Anxiety by Elias Wong

“Math anxiety affects different not only students but also people in different ways. It’s important to be familiar with the thoughts you have about yourself and the situation when you encounter math. If you are aware of unrealistic or irrational thoughts you can work to replace those thoughts with more positive and realistic ones.”

Wong writes about the phenomenon known as “math anxiety.” This term is used to describe many people’s hatred or fear of math- they feel that they are incapable of doing it. This anxiety is caused mainly by students’ negative experiences in math class, which makes them believe they cannot do well. Wong explains that some people have brains geared towards math and others do not, but this should not stop people from trying to overcome their math anxiety. Through review and practice of basic mathematical skills, students can overcome them and even excel at math. 

“We see that math is not an obscure subject reserved for some pretentious intellectual nobility. Though we may not be aware of it, mathematics is embedded into many different aspects of our lives and our world — and by understanding it deeply, we may just gain a greater understanding of ourselves.”

Similar to some of the previous essays, Ali’s essay explains the importance of math. Interestingly, he tells a story of the life of a person name Kyle. He goes through the typical stages of life and enjoys typical human hobbies, including Rubik’s cube solving. Throughout this “Kyle’s” entire life, he performed the role of a mathematician in various ways. Ali explains that math is much more prevalent in our lives than we think, and by understanding it, we can better understand ourselves. 

Writing Prompts on Essays about Math

Math is a controversial subject that many people either passionately adore or despise. In this essay, reflect on your feelings towards math, and state your position on the topic. Then, give insights and reasons as to why you feel this way. Perhaps this subject comes easily to you, or perhaps it’s a subject that you find pretty challenging. For an insightful and compelling essay, you can include personal anecdotes to relate to your argument. 

It is well-known that many people despise math. In this essay, discuss why so many people do not enjoy maths and struggle with this subject in school. For a compelling essay, gather interview data and statistics to support your arguments. You could include different sections correlating to why people do not enjoy this subject.

In this essay, begin by reading articles and essays about the importance of studying math. Then, write about the different ways that having proficient math skills can help you later in life. Next, use real-life examples of where maths is necessary, such as banking, shopping, planning holidays, and more! For an engaging essay, use some anecdotes from your experiences of using math in your daily life.

Many people have said that math is essential for the future and that you shouldn’t take a math class for granted. However, many also say that only a basic understanding of math is essential; the rest depends on one’s career. Is it essential to learn calculus and trigonometry? Choose your position and back up your claim with evidence. 

Prasanna’s essay lists down just a few applications math has in our daily lives. For this essay, you can choose any activity, whether running, painting, or playing video games, and explain how math is used there. Then, write about mathematical concepts related to your chosen activity and explain how they are used. Finally, be sure to link it back to the importance of math, as this is essentially the topic around which your essay is based. 

If you are interested in learning more, check out our essay writing tips !

For help with your essays, check out our round-up of the best essay checkers

Martin is an avid writer specializing in editing and proofreading. He also enjoys literary analysis and writing about food and travel.

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To enable every aspiring TJ student to present their best self

See previous problem solving essay prompts here, check out our feature in wtop, preparing for the essays has always been a vague, sometimes debated concept. one easy way to prepare for the problem solving (also called math based, science based and ethics) essays is to try the previous years'. each one should be done in an hour end to end. the most important thing to do in these is to explain and show your thoughts. though the right answer is a great, being able to show your thoughts is so much more important. remember to write them without the use of a calculator and an application that does not have spell check (ie. notepad).

It is  5:30 AM, and as a helicopter pilot you’ve just been told there’s an injured man on a boat you need to get to a hospital. The boat is going towards where you are at 10 mph, but is currently 400 miles away. You need to get him as soon as possible, but you only have 6600 lbs of fuel in your helicopter, which burns 1200 lbs per hour, and always travels at 150 mph. Also, you need to account for 30 min of fuel spent hovering over the boat to get the man into the helicopter, and 1 extra hour of fuel due to helicopter standards. When do you leave and why?

You are a researcher in the Falkland Islands You’ve been asked to estimate the penguin population. The current population is 1000 penguins. 500 pairs produce one chick per pair per year. At the end of each year there is a loss of 20% of the total population. Figure out how many penguins there will be at the end of seven years. Develop a formula that solves the problem.

When designing a stairway, an architect can use the riser-tread formula 2h +d = 25, where h is the riser height, in inches, and d is the tread depth, in inches. For any given stairway, the riser heights are the same and the tread depths are the same for all steps in that stairway.

The number of steps in a stairway is the number of its risers. For example, there are 5 steps in the stairway in the figure above. The total rise of a stairway is the sum of the riser heights as shown in the figure. 

An architect wants to use the riser-tread formula to design a stairway with a total rise of 9 feet, a riser height between 7 and 8 inches, and an odd number of steps. With the architect’s constraints, which of the following must be the tread depth, in inches, of the stairway?

To see the figure, click here

The Chesapeake Bay holds about 18 trillion gallons of water. The Chesapeake Bay covers about 4,480 square miles or about 3,000,000 acres. Each acre can hold 750,000 oysters and each oyster is capable of filtering 50 gallons of water per day. Can there be enough oysters to filter the Chesapeake Bay in one day?

***Please not that these are most likely not exactly how the prompts were worded. We searched the internet to compile these and cross referenced to find the most consistent version. These are great practice but not guaranteed the prompts verbatim***

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Problem Solving: Use of Math in Our Everyday Life

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