math application essay

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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Sat / act prep online guides and tips, the complete ib extended essay guide: examples, topics, and ideas.

International Baccalaureate (IB)

IB students around the globe fear writing the Extended Essay, but it doesn't have to be a source of stress! In this article, I'll get you excited about writing your Extended Essay and provide you with the resources you need to get an A on it.

If you're reading this article, I'm going to assume you're an IB student getting ready to write your Extended Essay. If you're looking at this as a potential future IB student, I recommend reading our introductory IB articles first, including our guide to what the IB program is and our full coverage of the IB curriculum .

IB Extended Essay: Why Should You Trust My Advice?

I myself am a recipient of an IB Diploma, and I happened to receive an A on my IB Extended Essay. Don't believe me? The proof is in the IBO pudding:

If you're confused by what this report means, EE is short for Extended Essay , and English A1 is the subject that my Extended Essay topic coordinated with. In layman's terms, my IB Diploma was graded in May 2010, I wrote my Extended Essay in the English A1 category, and I received an A grade on it.

What Is the Extended Essay in the IB Diploma Programme?

The IB Extended Essay, or EE , is a mini-thesis you write under the supervision of an IB advisor (an IB teacher at your school), which counts toward your IB Diploma (learn more about the major IB Diploma requirements in our guide) . I will explain exactly how the EE affects your Diploma later in this article.

For the Extended Essay, you will choose a research question as a topic, conduct the research independently, then write an essay on your findings . The essay itself is a long one—although there's a cap of 4,000 words, most successful essays get very close to this limit.

Keep in mind that the IB requires this essay to be a "formal piece of academic writing," meaning you'll have to do outside research and cite additional sources.

The IB Extended Essay must include the following:

• A title page
• Contents page
• Introduction
• Body of the essay
• References and bibliography

Additionally, your research topic must fall into one of the six approved DP categories , or IB subject groups, which are as follows:

• Group 1: Studies in Language and Literature
• Group 2: Language Acquisition
• Group 3: Individuals and Societies
• Group 4: Sciences
• Group 5: Mathematics
• Group 6: The Arts

Once you figure out your category and have identified a potential research topic, it's time to pick your advisor, who is normally an IB teacher at your school (though you can also find one online ). This person will help direct your research, and they'll conduct the reflection sessions you'll have to do as part of your Extended Essay.

As of 2018, the IB requires a "reflection process" as part of your EE supervision process. To fulfill this requirement, you have to meet at least three times with your supervisor in what the IB calls "reflection sessions." These meetings are not only mandatory but are also part of the formal assessment of the EE and your research methods.

According to the IB, the purpose of these meetings is to "provide an opportunity for students to reflect on their engagement with the research process." Basically, these meetings give your supervisor the opportunity to offer feedback, push you to think differently, and encourage you to evaluate your research process.

The final reflection session is called the viva voce, and it's a short 10- to 15-minute interview between you and your advisor. This happens at the very end of the EE process, and it's designed to help your advisor write their report, which factors into your EE grade.

Here are the topics covered in your viva voce :

• A check on plagiarism and malpractice
• Your reflection on your project's successes and difficulties
• Your reflection on what you've learned during the EE process

Your completed Extended Essay, along with your supervisor's report, will then be sent to the IB to be graded. We'll cover the assessment criteria in just a moment.

We'll help you learn how to have those "lightbulb" moments...even on test day!  

What Should You Write About in Your IB Extended Essay?

You can technically write about anything, so long as it falls within one of the approved categories listed above.

It's best to choose a topic that matches one of the IB courses , (such as Theatre, Film, Spanish, French, Math, Biology, etc.), which shouldn't be difficult because there are so many class subjects.

Here is a range of sample topics with the attached extended essay:

• Biology: The Effect of Age and Gender on the Photoreceptor Cells in the Human Retina
• Chemistry: How Does Reflux Time Affect the Yield and Purity of Ethyl Aminobenzoate (Benzocaine), and How Effective is Recrystallisation as a Purification Technique for This Compound?
• English: An Exploration of Jane Austen's Use of the Outdoors in Emma
• Geography: The Effect of Location on the Educational Attainment of Indigenous Secondary Students in Queensland, Australia
• Math: Alhazen's Billiard Problem
• Visual Arts: Can Luc Tuymans Be Classified as a Political Painter?

You can see from how varied the topics are that you have a lot of freedom when it comes to picking a topic . So how do you pick when the options are limitless?

How to Write a Stellar IB Extended Essay: 6 Essential Tips

Below are six key tips to keep in mind as you work on your Extended Essay for the IB DP. Follow these and you're sure to get an A!

#1: Write About Something You Enjoy

You can't expect to write a compelling essay if you're not a fan of the topic on which you're writing. For example, I just love British theatre and ended up writing my Extended Essay on a revolution in post-WWII British theatre. (Yes, I'm definitely a #TheatreNerd.)

I really encourage anyone who pursues an IB Diploma to take the Extended Essay seriously. I was fortunate enough to receive a full-tuition merit scholarship to USC's School of Dramatic Arts program. In my interview for the scholarship, I spoke passionately about my Extended Essay; thus, I genuinely think my Extended Essay helped me get my scholarship.

But how do you find a topic you're passionate about? Start by thinking about which classes you enjoy the most and why . Do you like math classes because you like to solve problems? Or do you enjoy English because you like to analyze literary texts?

Keep in mind that there's no right or wrong answer when it comes to choosing your Extended Essay topic. You're not more likely to get high marks because you're writing about science, just like you're not doomed to failure because you've chosen to tackle the social sciences. The quality of what you produce—not the field you choose to research within—will determine your grade.

Once you've figured out your category, you should brainstorm more specific topics by putting pen to paper . What was your favorite chapter you learned in that class? Was it astrophysics or mechanics? What did you like about that specific chapter? Is there something you want to learn more about? I recommend spending a few hours on this type of brainstorming.

One last note: if you're truly stumped on what to research, pick a topic that will help you in your future major or career . That way you can use your Extended Essay as a talking point in your college essays (and it will prepare you for your studies to come too!).

#2: Select a Topic That Is Neither Too Broad nor Too Narrow

There's a fine line between broad and narrow. You need to write about something specific, but not so specific that you can't write 4,000 words on it.

You can't write about WWII because that would be a book's worth of material. You also don't want to write about what type of soup prisoners of war received behind enemy lines, because you probably won’t be able to come up with 4,000 words of material about it. However, you could possibly write about how the conditions in German POW camps—and the rations provided—were directly affected by the Nazis' successes and failures on the front, including the use of captured factories and prison labor in Eastern Europe to increase production. WWII military history might be a little overdone, but you get my point.

If you're really stuck trying to pinpoint a not-too-broad-or-too-narrow topic, I suggest trying to brainstorm a topic that uses a comparison. Once you begin looking through the list of sample essays below, you'll notice that many use comparisons to formulate their main arguments.

I also used a comparison in my EE, contrasting Harold Pinter's Party Time with John Osborne's Look Back in Anger in order to show a transition in British theatre. Topics with comparisons of two to three plays, books, and so on tend to be the sweet spot. You can analyze each item and then compare them with one another after doing some in-depth analysis of each individually. The ways these items compare and contrast will end up forming the thesis of your essay!

When choosing a comparative topic, the key is that the comparison should be significant. I compared two plays to illustrate the transition in British theatre, but you could compare the ways different regional dialects affect people's job prospects or how different temperatures may or may not affect the mating patterns of lightning bugs. The point here is that comparisons not only help you limit your topic, but they also help you build your argument.

Comparisons are not the only way to get a grade-A EE, though. If after brainstorming, you pick a non-comparison-based topic and are still unsure whether your topic is too broad or narrow, spend about 30 minutes doing some basic research and see how much material is out there.

If there are more than 1,000 books, articles, or documentaries out there on that exact topic, it may be too broad. But if there are only two books that have any connection to your topic, it may be too narrow. If you're still unsure, ask your advisor—it's what they're there for! Speaking of advisors...

Don't get stuck with a narrow topic!

#3: Choose an Advisor Who Is Familiar With Your Topic

If you're not certain of who you would like to be your advisor, create a list of your top three choices. Next, write down the pros and cons of each possibility (I know this sounds tedious, but it really helps!).

For example, Mr. Green is my favorite teacher and we get along really well, but he teaches English. For my EE, I want to conduct an experiment that compares the efficiency of American electric cars with foreign electric cars.

I had Ms. White a year ago. She teaches physics and enjoyed having me in her class. Unlike Mr. Green, Ms. White could help me design my experiment.

Based on my topic and what I need from my advisor, Ms. White would be a better fit for me than would Mr. Green (even though I like him a lot).

The moral of my story is this: do not just ask your favorite teacher to be your advisor . They might be a hindrance to you if they teach another subject. For example, I would not recommend asking your biology teacher to guide you in writing an English literature-based EE.

There can, of course, be exceptions to this rule. If you have a teacher who's passionate and knowledgeable about your topic (as my English teacher was about my theatre topic), you could ask that instructor. Consider all your options before you do this. There was no theatre teacher at my high school, so I couldn't find a theatre-specific advisor, but I chose the next best thing.

Before you approach a teacher to serve as your advisor, check with your high school to see what requirements they have for this process. Some IB high schools require your IB Extended Essay advisor to sign an Agreement Form , for instance.

Make sure that you ask your IB coordinator whether there is any required paperwork to fill out. If your school needs a specific form signed, bring it with you when you ask your teacher to be your EE advisor.

#4: Pick an Advisor Who Will Push You to Be Your Best

Some teachers might just take on students because they have to and aren't very passionate about reading drafts, only giving you minimal feedback. Choose a teacher who will take the time to read several drafts of your essay and give you extensive notes. I would not have gotten my A without being pushed to make my Extended Essay draft better.

Ask a teacher that you have experience with through class or an extracurricular activity. Do not ask a teacher that you have absolutely no connection to. If a teacher already knows you, that means they already know your strengths and weaknesses, so they know what to look for, where you need to improve, and how to encourage your best work.

Also, don't forget that your supervisor's assessment is part of your overall EE score . If you're meeting with someone who pushes you to do better—and you actually take their advice—they'll have more impressive things to say about you than a supervisor who doesn't know you well and isn't heavily involved in your research process.

Be aware that the IB only allows advisors to make suggestions and give constructive criticism. Your teacher cannot actually help you write your EE. The IB recommends that the supervisor spends approximately two to three hours in total with the candidate discussing the EE.

#5: Make Sure Your Essay Has a Clear Structure and Flow

The IB likes structure. Your EE needs a clear introduction (which should be one to two double-spaced pages), research question/focus (i.e., what you're investigating), a body, and a conclusion (about one double-spaced page). An essay with unclear organization will be graded poorly.

The body of your EE should make up the bulk of the essay. It should be about eight to 18 pages long (again, depending on your topic). Your body can be split into multiple parts. For example, if you were doing a comparison, you might have one third of your body as Novel A Analysis, another third as Novel B Analysis, and the final third as your comparison of Novels A and B.

If you're conducting an experiment or analyzing data, such as in this EE , your EE body should have a clear structure that aligns with the scientific method ; you should state the research question, discuss your method, present the data, analyze the data, explain any uncertainties, and draw a conclusion and/or evaluate the success of the experiment.

#6: Start Writing Sooner Rather Than Later!

You will not be able to crank out a 4,000-word essay in just a week and get an A on it. You'll be reading many, many articles (and, depending on your topic, possibly books and plays as well!). As such, it's imperative that you start your research as soon as possible.

Each school has a slightly different deadline for the Extended Essay. Some schools want them as soon as November of your senior year; others will take them as late as February. Your school will tell you what your deadline is. If they haven't mentioned it by February of your junior year, ask your IB coordinator about it.

Some high schools will provide you with a timeline of when you need to come up with a topic, when you need to meet with your advisor, and when certain drafts are due. Not all schools do this. Ask your IB coordinator if you are unsure whether you are on a specific timeline.

Below is my recommended EE timeline. While it's earlier than most schools, it'll save you a ton of heartache (trust me, I remember how hard this process was!):

• January/February of Junior Year: Come up with your final research topic (or at least your top three options).
• February of Junior Year: Approach a teacher about being your EE advisor. If they decline, keep asking others until you find one. See my notes above on how to pick an EE advisor.
• April/May of Junior Year: Submit an outline of your EE and a bibliography of potential research sources (I recommend at least seven to 10) to your EE advisor. Meet with your EE advisor to discuss your outline.
• Summer Between Junior and Senior Year: Complete your first full draft over the summer between your junior and senior year. I know, I know—no one wants to work during the summer, but trust me—this will save you so much stress come fall when you are busy with college applications and other internal assessments for your IB classes. You will want to have this first full draft done because you will want to complete a couple of draft cycles as you likely won't be able to get everything you want to say into 4,000 articulate words on the first attempt. Try to get this first draft into the best possible shape so you don't have to work on too many revisions during the school year on top of your homework, college applications, and extracurriculars.
• August/September of Senior Year: Turn in your first draft of your EE to your advisor and receive feedback. Work on incorporating their feedback into your essay. If they have a lot of suggestions for improvement, ask if they will read one more draft before the final draft.
• September/October of Senior Year: Submit the second draft of your EE to your advisor (if necessary) and look at their feedback. Work on creating the best possible final draft.
• November-February of Senior Year: Schedule your viva voce. Submit two copies of your final draft to your school to be sent off to the IB. You likely will not get your grade until after you graduate.

Remember that in the middle of these milestones, you'll need to schedule two other reflection sessions with your advisor . (Your teachers will actually take notes on these sessions on a form like this one , which then gets submitted to the IB.)

I recommend doing them when you get feedback on your drafts, but these meetings will ultimately be up to your supervisor. Just don't forget to do them!

The early bird DOES get the worm!

How Is the IB Extended Essay Graded?

Extended Essays are graded by examiners appointed by the IB on a scale of 0 to 34 . You'll be graded on five criteria, each with its own set of points. You can learn more about how EE scoring works by reading the IB guide to extended essays .

• Criterion A: Focus and Method (6 points maximum)
• Criterion B: Knowledge and Understanding (6 points maximum)
• Criterion C: Critical Thinking (12 points maximum)
• Criterion D: Presentation (4 points maximum)
• Criterion E: Engagement (6 points maximum)

How well you do on each of these criteria will determine the final letter grade you get for your EE. You must earn at least a D to be eligible to receive your IB Diploma.

Although each criterion has a point value, the IB explicitly states that graders are not converting point totals into grades; instead, they're using qualitative grade descriptors to determine the final grade of your Extended Essay . Grade descriptors are on pages 102-103 of this document .

Here's a rough estimate of how these different point values translate to letter grades based on previous scoring methods for the EE. This is just an estimate —you should read and understand the grade descriptors so you know exactly what the scorers are looking for.

Here is the breakdown of EE scores (from the May 2021 bulletin):

How Does the Extended Essay Grade Affect Your IB Diploma?

The Extended Essay grade is combined with your TOK (Theory of Knowledge) grade to determine how many points you get toward your IB Diploma.

To learn about Theory of Knowledge or how many points you need to receive an IB Diploma, read our complete guide to the IB program and our guide to the IB Diploma requirements .

This diagram shows how the two scores are combined to determine how many points you receive for your IB diploma (3 being the most, 0 being the least). In order to get your IB Diploma, you have to earn 24 points across both categories (the TOK and EE). The highest score anyone can earn is 45 points.

Let's say you get an A on your EE and a B on TOK. You will get 3 points toward your Diploma. As of 2014, a student who scores an E on either the extended essay or TOK essay will not be eligible to receive an IB Diploma .

Prior to the class of 2010, a Diploma candidate could receive a failing grade in either the Extended Essay or Theory of Knowledge and still be awarded a Diploma, but this is no longer true.

Figuring out how you're assessed can be a little tricky. Luckily, the IB breaks everything down here in this document . (The assessment information begins on page 219.)

40+ Sample Extended Essays for the IB Diploma Programme

In case you want a little more guidance on how to get an A on your EE, here are over 40 excellent (grade A) sample extended essays for your reading pleasure. Essays are grouped by IB subject.

• Business Management 1
• Chemistry 1
• Chemistry 2
• Chemistry 3
• Chemistry 4
• Chemistry 5
• Chemistry 6
• Chemistry 7
• Computer Science 1
• Economics 1
• Design Technology 1
• Design Technology 2
• Environmental Systems and Societies 1
• Geography 1
• Geography 2
• Geography 3
• Geography 4
• Geography 5
• Geography 6
• Literature and Performance 1
• Mathematics 1
• Mathematics 2
• Mathematics 3
• Mathematics 4
• Mathematics 5
• Philosophy 1
• Philosophy 2
• Philosophy 3
• Philosophy 4
• Philosophy 5
• Psychology 1
• Psychology 2
• Psychology 3
• Psychology 4
• Psychology 5
• Social and Cultural Anthropology 1
• Social and Cultural Anthropology 2
• Social and Cultural Anthropology 3
• Sports, Exercise and Health Science 1
• Sports, Exercise and Health Science 2
• Visual Arts 1
• Visual Arts 2
• Visual Arts 3
• Visual Arts 4
• Visual Arts 5
• World Religion 1
• World Religion 2
• World Religion 3

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College admissions

Course: college admissions   >   unit 4.

• Writing a strong college admissions essay
• Avoiding common admissions essay mistakes
• Brainstorming tips for your college essay
• How formal should the tone of your college essay be?
• Taking your college essay to the next level
• Sample essay 1 with admissions feedback

Sample essay 2 with admissions feedback

• Student story: Admissions essay about a formative experience
• Student story: Admissions essay about personal identity
• Student story: Admissions essay about community impact
• Student story: Admissions essay about a past mistake
• Student story: Admissions essay about a meaningful poem
• Writing tips and techniques for your college essay

Introduction

Sample essay 2, feedback from admissions.

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About You: the writing section

In your Mathcamp application, there will be an About You section: a chance for you to speak directly to the admissions committee about yourself and what motivates you to apply.

In a nutshell, we would like you to tell us a little more about yourself, your interest in math, and why you want to come to Mathcamp. Most applicants answer in a few paragraphs (typically 400–1000 words), but there is neither a minimum nor a maximum word count: you can feel free to express your ideas at their natural length.

The format is completely up to you: it can be a series of short answers, or one overarching narrative. In particular, this doesn't have to be a formal essay: in fact, we'll get to know you better if you write in your natural conversational voice. You could imagine you're talking about yourself to a friend. You're also welcome to get creative if you'd like: we've received lists, poems, puzzles, short stories, plays, letters – all kinds of things. Just use this space to introduce yourself to us.

In particular, we want to hear from you , not from ChatGPT. See our Policy on Getting Help for more details.

Below are some examples of things you could talk about (but don't feel obliged to respond to every prompt!).

We are particularly interested in hearing about...

• How do you develop your interest in math?
• Why do you want to come to Mathcamp? Do you have any specific goals, mathematical or otherwise?
• What's something you do not because it's required of you, but just for fun?

You might also cover one or more of the following:

• There's a very vibrant life at Mathcamp outside of the classroom ! What camp activity would you like to see run (or even run yourself)?
• Tell us about a really beautiful or compelling mathematical idea that you have encountered. What's so amazing about it?
• As we're getting to know you, is there anything you think it's important for us to keep in mind? For example: if our picture wouldn't be complete without knowing about an aspect of your identity, or an animating passion that has been central to your life, then please tell us about it.

It goes without saying that there are no "right answers" to any of these questions: they're just here to help the admissions committee to understand the role that Mathcamp might play in your life, and the role that you might play in the Mathcamp community. That said, feel free to write about anything that could give us some sense of who you are – as a person, a mathematician, and someone with whom we might get to spend five weeks this summer.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

1— Mathematics as a Gateway to Student Success

DALE PARNELL

Oregon State University

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

2— Market Launch

ROL FESSENDEN

L. L. Bean, Inc.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

3— Integrating Vocational and Academic Education

THOMAS BAILEY

Columbia University

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

Rethinking Vocational and Academic Education

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

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

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

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

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

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

Innovative Examples

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

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

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

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

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

Joining Work and Learning

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

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

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

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

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

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

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

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

4— The Importance of Workplace and Everyday Mathematics

JEAN E. TAYLOR

Rutgers University

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

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

What Should Students Learn for Today's World?

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

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

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

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

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

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

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

Workplace and Everyday Settings as a Way of Making Sense

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

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

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

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

Concepts or Procedures?

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

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

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

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

An Example: The Student/Professor Problem

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

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

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

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

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

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

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

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

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

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

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

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

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

5— Working with Algebra

DANIEL CHAZAN

Michigan State University

SANDRA CALLIS BETHELL

Holt High School

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

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

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

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

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

Our Teaching and Issues it Raised

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

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

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

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

An Approach to School Algebra

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

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

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

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

The "Sponsor" Project

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

Using These Questions

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

TABLE 5-1: Questions to ask in the workplace

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

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

Some Examples of Students' Work

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

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

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

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

TABLE 5-2: Cost of carpet worksheet

FIGURE 5-1: Pregnancy wheel

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Emergency Calls

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

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

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

Mathematical Analysis

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

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

FIGURE 1: Distribution of Arrow's response times

FIGURE 2: Distribution of Metro's response times

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

FIGURE 3: Arrow response times by time of day

FIGURE 4: Metro response times by time of day

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

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

Back-of-the-Envelope Estimates

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

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

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

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

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

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

The examples are solved separately below.

Grading Homework

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

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

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

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

The New York Times

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

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

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

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

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

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

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

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

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

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

Scheduling Elevators

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

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

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

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

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

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

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

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

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

Scenario One

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

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

TABLE 1: Elevator round-trip time, Scenario one

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

Scenario Two

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

TABLE 2: Elevator round-trip times, Scenario two

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

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

Scenario Three

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

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

TABLE 3: Elevator round-trip times, Scenario three

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Heating-Degree-Days

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

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

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

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

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

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

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

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

FIGURE 1: Daily heating-degree-days

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Essays 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. 

You might also be interested in our ordinal number explainer.

“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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College Admission Glossary: Learn the Lingo

Find the right college for you..

What does admission mean? What is a transcript? What's the difference between early action and early decision? When applying to college, you're bound to come across unfamiliar collegiate terms. This glossary can help you make sense of all the college terms you're sorting through.

A standardized college admission test. It features four main sections: English, math, reading and science — and an optional essay section.

Admissions Process

The entire process you go through to get into college. To define admissions, you need to include a number of components. The process starts early in high school as you begin building your GPA and ends when you move into your dorm. Your research, college visits, applications, letters of recommendation, high school transcripts, and admissions essays are all important parts of the admissions meaning.

Admissions Decisions

The decisions made by the college admissions committee about which applicants receive acceptance letters and which applicants do not. Early decisions are available to students who submit their applications within each school's early decision timeline. See "Early Decision (ED)" for further information.

Admission Tests

Also known as college entrance exams, these are tests designed to measure students' skills and help colleges evaluate how ready students are for college-level work. The ACT and the College Board's SAT are two standardized admission tests used in the United States. The word "standardized" means that the test measures the same thing in the same way for everyone who takes it. Read more about admission tests .

Articulation Agreement

An agreement between two-year and four-year colleges that makes it easier to transfer credits between them. It spells out which courses count for degree credit and the grades you need to earn to get credit.

Bachelor's Degree

When you complete the required courses and earn enough credits, typically after four years in college, you will earn a BA or bachelor's degree. Depending upon your major, you may earn a bachelor of arts (BA), bachelor of science (BS), or bachelor of fine arts (BFA).

Candidates Reply Date Agreement (CRDA)

An agreement many colleges follow that gives applicants until May 1 to accept or decline offers of admission. This deadline gives students time to get responses from most of the colleges they have applied to before deciding on one.

A measurement of how your academic achievement in high school compares with that of other students in your grade. Your class ranking is primarily based on your GPA. Some schools calculate class rank differently than others, so check with your school officials for information on how your school calculates GPA.

Coalition Application

A standard application form accepted by members of the Coalition for College. You can use this application to apply to any of the more than 90 colleges and universities that are members of the Coalition.

College Application Essay

An essay that a college requires students to write and submit as part of their application. Some colleges offer applicants specific questions to answer, while others simply ask applicants to write about themselves. Colleges may refer to this as a “personal statement.” Learn more about college application essays .

College Credit

What you get when you successfully complete a college-level course. You need a certain number of credits to graduate with a degree. Colleges may also grant credit for scores on exams, such as those offered by the College Board’s AP Program® and CLEP. Learn more about AP and CLEP . 

College Financial Aid Offer

An offer of financial assistance to those who receive acceptance to a college. The offer includes all the monetary assistance available to you to pay for college. Also called an award letter, a financial aid award letter, or simply an offer, your letter will include the need-based assistance you qualify to receive based on your FAFSA® (Free Application for Federal Student Aid) and any other required forms such as the CSS Profile or an institutional form.

Common Application

A standard application form accepted by all colleges that are members of the Common Application association. The definition of the Common Application is synonymous with college application. You can fill out the Common App once and submit it to any one or several of the nearly 700 colleges that accept it. Go to the Common Application .

Deferred Admission

Permission from a college that has accepted you to postpone enrolling in the college. The postponement is usually for up to one year. Often, a college requires a good-faith deposit to hold your place for the deferment period. Additionally, your college may ask you to account for your experiences during deferment.

Demonstrated Interest

An action that demonstrates you have a sincere interest in attending a particular college. Many admissions committees consider this factor in making their decisions. Some of the ways you can demonstrate an interest in attending include a campus visit or virtual tour participation, having an initial interview, conversations with admission representatives, and applying for early admission.

Early Action (EA)

An option to submit an application to your first-choice college before the regular deadline. When you apply early decision, you get an admission decision earlier than usual. Early decision plans are binding. You agree to enroll in the college immediately if admitted and offered a financial aid package that meets your needs. Some colleges have an early decision option called ED II, which has a later application deadline than their regular ED plan. Learn more about applying early .

Early Decision (ED)

The Free Application for Federal Student Aid (FAFSA). Everyone planning to attend college should fill in and submit a FAFSA prior to their first year. It determines if you qualify for financial assistance with tuition so you can attend the school of your choice. Your college offer may be based on your FAFSA results.

Financial Aid

Money given or loaned to you to help pay for college. Financial aid can come from federal and state governments, colleges, and private organizations. It might also include work-study assistance. Learn more about financial aid .

Grade Point Average (GPA)

A number that shows overall academic performance. It's computed by assigning a point value to each grade you earn. It is also a key factor in determining your class rank.

Legacy Applicant

A college applicant with a relative (usually a parent or grandparent) who graduated from that college. Some colleges give preference to legacy applicants (also called "legacies").

Need-Blind Admission

A policy of making admission decisions without considering the financial circumstances of applicants. Colleges that use this policy may or may not offer enough financial aid to meet a student's full need.

Open Admission

A policy of accepting any high school graduate, no matter what his or her grades are, until all spaces in the incoming class are filled. Almost all two-year community colleges have an open-admission policy. However, a college with a general open-admission policy may have admission requirements for certain programs.

Placement Tests

Tests that measure the academic skills needed for college-level work. They cover reading, writing, math, and sometimes other subjects. Placement test results help determine what courses you are ready for and whether you would benefit from remedial classes. They can also determine whether you need to take a basic course. Read more about placement tests . 

Priority Date or Deadline

The date by which your application—whether it's for college admission, student housing, or financial aid—must be received to be given the strongest consideration.

The college official who registers students. The registrar may also be responsible for keeping permanent records and maintaining your student file.

Rolling Admission

An admission policy of considering each application as soon as all required information (such as high school records and test scores) has been received, rather than setting an application deadline and reviewing applications in a batch. Colleges that use a rolling admission policy usually notify applicants of admission decisions quickly.

College Board’s standardized college admission test. It features three main sections: math, reading and writing, which includes a written essay. Learn more about the SAT .

Sophomore Standing

The status of a second-year student. A college may grant sophomore standing to an incoming first-year student if they have earned college credits through courses, exams, or other programs at a previous school.

The official record of your coursework at a school or college. Your high school transcript is usually required for college admission, and for some financial aid packages or scholarship applications.

Transfer Student

A student who enrolls in a college after having attended another college. Before transferring, you should check with your current and future colleges to find out which credits will transfer.

Undergraduate

A college student who is working toward an associate degree or a bachelor's degree.

Universal College Application

A standard application form accepted by all colleges that are Universal College Application (UCA) members. Established in 2007, this application offers shortcuts—such as no recommendation letter requirement—that may help you complete your package sooner. However, not all colleges accept it, so check with your school to make sure. Go to the Universal College Application .

Waiting List

The list of applicants who may be admitted to a college if space becomes available. Colleges wait to hear if all the students they accepted decide to attend. If students don't enroll and there are empty spots, a college may fill them with students who are on the waiting list. Learn more about waiting lists .

Weighted Grade Point Average (GPA)

A grade point average that's calculated using a system that assigns a higher point value to grades in certain classes, typically more difficult ones. For example, some high schools assign the value of 5.0 (instead of the standard 4.0) for an A earned in an AP class.

Virtual College

Online college classes. Some colleges are entirely virtual and do not maintain an actual campus, while others offer online and in-person courses. While virtual college classes are often more convenient because they allow you to set your own schedule, some subjects, such as science labs, require hands-on participation that you cannot do online.

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The Summer Academy for Math and Science (SAMS) Application 2023 Essay Prompt Guide

Requirements: 1 essay of 300-500 words, 1 essay of up to 1,000 words

Deadline: March 15, 2023

An essay is required for the following prompt (300-500 words):

What do you hope to gain from participating in a carnegie mellon pre-college program.

This prompt is a straightforward classic in the admissions game. In other words: Why are you applying? What do you hope to get out of your SAMS experience? We recommend starting with a piece of paper (if you’re old school) or a blank document and writing down everything that comes to mind. Don’t worry about editing; you can do that later. Next, go to CMU’s webpage for the SAMS program and do your due diligence to find out what you can expect. The more information you can incorporate into your response, the more specific and authentic your essay will be. Show admissions you’re a serious applicant making moves to achieve your dreams (and SAMS is a part of your journey).

In addition, respond to one of the two following prompts (no more than 1000 words):

It is often said that adversity builds character, and frequently the lessons we take from encountered obstacles can build a foundation to later success. carefully recount a time that you faced a very specific challenge. name the setback or failure, and describe how it impacted and influenced your values. how did this experience affect you what were your lessons learned.

Essays about overcoming obstacles are really an opportunity to speak about learning experiences, growth, and resilience. The obstacle you choose to explore can vary widely in nature; it can be as serious as being tormented by bullies, as ingrained as the financial issues that have plagued your family for years, or as seemingly pedestrian as a mistake that cost you a tip while waiting tables. While the possibilities are almost endless, students should be careful not to choose challenges that may seem trite (the inability to achieve an A on an exam and/or secure tickets to that Billie Eilish concert) or that illustrate a lapse in good judgment (that time you crashed a golf cart or ate 20 mozzarella sticks in one sitting). If you can isolate an incident of trial in your life and illustrate how you learned from it, you will show admissions the kind of thoughtful, incisive person you are. 

In 1900 Carnegie Mellon’s founder, Andrew Carnegie, stated, “My Heart is in the Work.” Understanding that one of the University’s foundational pillars is diversity, equity, and inclusion, please relate and connect this quote to your desire to attend the Summer Academy for Math and Science (SAMS). Discuss your interest in diversity, equity, and inclusion (DEI) as it relates to STEM fields and/or your own experiences with inclusive education.  How will SAMS help you in your educational interests and career pursuits?

There’s a lot going on in this prompt, so let’s break it down. Admissions wants to know (1) how you connect Carnegie’s statement, “My Heart is in the Work,” to your interest in SAMS, (2) how you interact with diversity, equity, and inclusion as they relate to your own experiences in STEM, and (3) how SAMS will help you achieve your goals.

The good news is, since you have up to 1,000 words to work with, you have the space to dive deep into your responses to each facet of the prompt. A successful essay will address all three points and offer admissions a clear understanding of what’s important to you, what you’re working toward, and how you will interact with students from different backgrounds. You can start by freewriting or making a bulleted list of answers to the aforementioned questions to get the gears in motion. Since SAMS asks applicants to pen longform essays, you have a lot of writing and editing ahead of you, so be sure and set aside enough time to do this justice. As a wise person once said, “The best time to start was yesterday, the next best time is now.”

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12 Great University of California Essay Examples

What’s covered, essay #1: leadership, essay #2: creativity, essay #3: creativity, essay #4: creativity, essay #5: talent, essay #6: talent, essay #7: academic interest, essay #8: academic interest, essay #9: community, essay #10: community, essay #11: community, essay #12: community.

The University of California system is comprised of nine undergraduate universities, and is one of the most prestigious public school systems in the country. The UC schools have their own application system, and students must respond to four of eight personal insight questions in 350 words each. Every UC school you apply to receives the same application and essays, so it’s important that your responses accurately represent your personality and writing abilities. 

In this post, we’ll share some UC essay examples and go over what they did well and where they could improve. We will also point you to free resources you can use to improve your college essays. 

Please note: Looking at examples of real essays students have submitted to colleges can be very beneficial to get inspiration for your essays. You should never copy or plagiarize from these examples when writing your own essays. Colleges can tell when an essay isn’t genuine and will not view students favorably if they plagiarized. 

Read our guide to the UC personal insight questions for more tips on writing strong essays for each of the prompts.

Prompt: Describe an example of your leadership experience in which you have positively influenced others, helped resolve disputes, or contributed to group efforts over time. (350 words)

1400 lines of code. 6 weeks. 1 Pizza.

I believe pizza makers are the backbone of society. Without pizza, life as we know it would cease to exist. From a toddler’s birthday party to President Obama’s sporadic campaigning cravings, these 8 slices of pure goodness cleverly seep into every one of our lives; yet, we never talk about it. In a very cheesy way, I find representation in a pizza maker. 

The most perplexing section of physiology is deciphering electrocardiograms. According to our teacher, this was when most students hit their annual trough. We had textbooks and worksheets, but viewing printed rhythms and attempting to recognize them in real-time is about as straining as watching someone eat pizza crust-first. Furthermore, online simulators were vastly over-engineered, featuring complex interfaces foreign to high-school students.

Eventually, I realized the only way to pull myself out of the sauce was by creating my own tools. This was also the first year I took a programming course, so I decided to initiate a little hobbyist experiment by extrapolating knowledge from Computer Science and Physiology to code and share my own Electrocardiogram Simulator. To enhance my program, I went beyond the textbook and classroom by learning directly from Java API – the programmer’s Bible.

The algorithms I wrote not only simulated rhythms in real-time but also actively engaged with the user, allowing my classmates and I to obtain a comprehensive understanding of the curriculum. Little did I know that a small project born out of desperation would eventually become a tool adopted by my teacher to serve hundreds of students in the future.

Like pizza, people will reap the benefits of my app over and over again, and hardly anyone will know its maker. Being a leader doesn’t always mean standing at the front of rallies, giving speeches, and leading organizations. Yes, I have done all three, but this app taught me leaders are also found behind-the-scenes, solving problems in unimaginable ways and fulfilling the hidden, yet crucial niches of the world. 

1400 lines of code, and 6 weeks later, it’s time to order a pizza. 

What the Essay Did Well

This is a great essay because it is both engaging and informative. What exactly does it inform us about? The answer: the personality, work ethic, and achievements of this student (exactly what admissions officers want to hear about).

With regards to personality, the pizza through-line—which notably starts the essay, ends the essay, and carries us through the essay—speaks volumes about this student. They are admittedly “cheesy,” but they appear unabashedly themself. They own their goofiness. That being said, the student’s pizza connections are also fitting and smoothly advance their points—watching someone eat pizza crust-first is straining and pizza is an invention that hardly anyone can identify the maker of. 

While we learn about this student’s fun personality in this essay, we also learn about their work ethic. A student who takes the initiative to solve a problem that no one asked them to solve is the kind of student an admissions officer wants to admit. The phrase “I decided to initiate a little hobbyist experiment” alone tells us that this student is a curious go-getter.

Lastly, this student tells us about their achievements in the last two paragraphs. Not only did they take the initiative to create this program, but it was also successful. On top of that, it’s notable how this student’s accomplishments as a leader defy the traditional expectations people have for leaders. The student’s ability to demonstrate their untraditional leadership path is an achievement in itself that sets the student apart form other applicants.

What Could Be Improved

This is a strong essay as is, but the one way this student could take it above and beyond would be to tell less and show more. To really highlight the student’s writing ability, the essay should  show the reader all the details it’s currently telling us. For example, these sentences primarily tell the reader what happened: “The most perplexing section of physiology is deciphering electrocardiograms. According to our teacher, this was when most students hit their annual trough.” 

Rewriting this sentence to show the reader the student’s impetus for creating their app could look like this: “When my teacher flashed the electrocardiogram on the screen, my once attentive physiology class became a sea of blank stares and furrowed brows.” This sentence still conveys the key details—student’s in the physiology class found electrocardiograms to be the hardest unit of the year—but it does so in a far more descriptive way. Implementing this exercise of rewriting sentences to show what happened throughout the piece would elevate the entire essay.

Prompt: Every person has a creative side, and it can be expressed in many ways: problem solving, original and innovative thinking, and artistically, to name a few. Describe how you express your creative side. (350 words)

For the past few years, participating in debate has been one of the foremost expressions of my creativity. Nothing is as electrifying as an Asian parliamentary-style debate. Each team is given only thirty minutes to prepare seven-minute speeches to either support or oppose the assigned motion. Given the immense time pressure, this is where my creativity shines most brightly.

To craft the most impactful and convincing argument, I have to consider the context of the motion, different stakeholders, the goals we want to achieve, the mechanisms to reach those goals, and so much more. I have to frame these arguments effectively and paint a compelling and cohesive world to sway my listeners to my side on both an emotional and logical level. For example, In a debate about the implementation of rice importation in the Philippines, I had to frequently switch between the macro perspective by discussing the broad economic implications of the policy and the micro perspective by painting a picture of the struggles that local rice farmers would experience when forcefully thrust into an increasingly competitive global economy. It’s a tough balancing act.

To add to the challenge, there is an opposing team on the other side of the room hell-bent on disproving everything I say. They generate equally plausible sounding arguments, and my mission is to react on the spot to dispel their viewpoints and build up our team’s case.

When two debate teams, both well-prepared and hungry for victory, face off and try to out-think one another, they clash to form a sixty-minute thunderstorm raining down fierce arguments and rebuttals. They fill up a room with unbelievable energy. After several years of debate, I have developed the capacity to still a room of fury and chaos with nothing but my words and wit.

Debate has been instrumental in shaping me into the person I am today. Because of debate, I have become a quicker and stronger thinker. Lightning quick on my feet, I am ready to thoroughly and passionately defend my beliefs at a moment’s notice.

This prompt is about creativity, though its wording emphasizes how students aren’t required to talk about typically-creative subjects. That said, it might take a bit more work and explanation (even creativity, one could say) to position a logical process as creative. This student’s main strength is the way they convince the reader that debate is creative.

First, they identify how “Asian parliamentary-style debate” differs from other forms of debate, emphasizing how time constraints necessitate the use of creativity. Then, they explain how both the argument’s content (the goals and solutions they outline) and the argument’s composition (the way they frame the argument) must be creatively orchestrated to be convincing. 

To drive home the point that debate is a creative process, this student provides an example of how they structured their argument about rice importation in the Philippines. This essay is successful because, after reading it, an admissions officer has no doubt that this student can combine logic and creativity to think intellectually.

One aspect of this essay that could be improved is the language use. Although there are some creative metaphors like the “sixty-minute thunderstorm raining down fierce arguments”, the essay is lacking the extra oomph and wow-factor that carefully chosen diction provides. In the second paragraph, the student repeats the phrase “I have to” three different times when stronger, more active verbs could have been used.

Essays should always reflect the student’s natural voice and shouldn’t sound like every word came straight out of a thesaurus, but that doesn’t mean they can’t incorporate a bit of colorful language. If this student took the time to go through their essay and ask themself if an overused word could be replaced with a more exciting one, it would make the essay much more interesting to read.

As I open the door to the Makerspace, I am greeted by a sea of cubicle-like machines and I watch eagerly, as one of them completes the final layer of my print.

Much like any scientific experiment, my countless failures in the Makerspace – hours spent designing a print, only to have it disintegrate – were my greatest teachers. I learned, the hard way, what types of shapes and patterns a 3D printer would play nice to. Then, drawing inspiration from the engineering method, I developed a system for myself – start with a solid foundation and add complexity with each iteration – a flourish here, a flying buttress there. 

But it wasn’t until the following summer, vacationing on a beach inundated with plastic, that the “aha” moment struck. In an era where capturing people’s attention in a split-second is everything, what better way to draw awareness to the plastic problem than with quirky 3D-printed products? By the time I had returned home, I had a business case on my hands and a desire to make my impact.

Equipped with vital skills from the advanced math-and-science courses I had taken in sophomore year, I began applying these to my growing business. Using my AP Chemistry analytical laboratory skills, I devised a simple water bath experiment to test the biodegradability claims of 3D-printer filaments from different manufacturers, guaranteeing that my products could serve as both a statement and play their part for our planet. The optimization techniques I had learned in AP Calculus were put to good use, as I determined the most space-efficient packaging for my products, reducing my dependence on unsustainable filler material. Even my designs were tweaked and riffed on to reflect my newfound maturity and keen eye for aesthetics.

My business is still going strong today, raising $1000 to date. I attribute this success to a fateful spark of creative inspiration, which has, and will, continue to inspire me to weave together multiple disciplines to address issues as endemic as the plastic problem. 

This essay begins with a simple, yet highly effective hook. It catches readers’ attention by only giving a hint about the essay’s main topic, and being a standalone paragraph makes it all the more intriguing. 

The next paragraph then begins with a seamless transition that ties back to the Makerspace. The essay goes on to show the writer’s creative side and how it has developed over time. Rather than directly stating “I am most creative when I am working on my business,” the writer tells the story of their creativity while working with 3-D printers and vacationing on the beach. 

It is the “aha” moment that perhaps responds to the prompt best. Here we get to see the writer create a new idea on the spot. The next two paragraphs then show the writer executing on their idea in great detail. Small and specific details, such as applying analytical laboratory skills from AP Chemistry, make the writer’s creativity come to life. 

From start to finish, this essay shows that the key to writing a stellar response to this prompt is to fill your writing with details and vivid imagery. 

The second to last paragraph of this essay focuses a bit too much on how the writer built their business. Though many of these details show the writer’s creativity in action, a few of them could be restated to make the connection to creativity clearer. The last sentences could be rewritten like so: 

Working on my business was where my creativity blossomed. In my workshop, optimization techniques that I learned in AP Calculus became something new — the basis for space-efficient packaging for my products that reduced my dependence on unsustainable filler material…

Profusely sweating after trying on what felt like a thousand different outfits, I collapsed on the floor in exasperation. The heaping pile of clothes on my bed stared me down in disdain; with ten minutes left to spare before the first day of seventh grade, I let go of my screaming thoughts and settled on the very first outfit I tried on: my favorite.

Donning a neon pink dress, that moment marked the first time I chose expression over fear. Being one of the few Asians in my grade, clothing was my source of disguise. I looked to the bold Stacy London of What Not to Wear for daily inspiration, but, in actuality, I dressed to conceal my uniqueness so I wouldn’t be noticed for my race. Wearing jeans and a t-shirt, I envied the popular girls who hiked their shorts up just a few inches higher than dress code allowed and flaunted Uggs decorated with plastic jewels, a statement that Stacy London would have viewed as heinous and my mother impractical. 

However, entering school that day and the days after, each compliment I received walking down the hallways slowly but surely broke down the armored shield. Morphing into an outlet to amplify my voice and creativity, dressing up soon became what I looked forward to each morning. I was awarded best dressed the year after that during my middle school graduation, a recognition most would scoff at. But, to me, that flimsy paper certificate was a warm embrace telling me that I was valued for my originality and expression. I was valued for my differences. 

Confidence was what I found and is now an essential accessory to every outfit I wear. Taking inspiration from vintage, simplistic silhouettes and Asian styles, I adorn my body’s canvas with a variety of fabrics and vibrant colors, no longer depriving it of the freedom to self expression and cultural exploration. I hope that my future will open new doors for me, closet doors included, at the University of California with opportunities to intertwine creativity with my identity even further.

Colorful language and emotion are conveyed powerfully in this essay, which is one of its key strengths. We can see this in the first paragraph, where the writer communicates that they were feeling searing judgment by using a metaphor: “the heaping pile of clothes on my bed stared me down.” The writer weaves other rich phrases into the essay — for example, “my screaming thoughts” — to show readers their emotions. All of these writing choices are much more moving than plainly stating “I was nervous.”

The essay moves on to tell a story that responds to the prompt in a unique way. While typical responses will be about a very direct example of expressing creativity, e.g. oil painting, this essay has a fittingly creative take on the prompt. The story also allows the writer to avoid a common pitfall — talking more about the means of being creative rather than how those means allow you to express yourself. In other words, make sure to avoid talking about the act of oil painting so much that your essay loses focus on what painting means to you.

The last sentence of the essay is one more part to emulate. “I hope that my future will open new doors for me, closet doors included…” is a well-crafted, flawlessly succinct metaphor that looks to the future while connecting the end of the essay to its beginning. The metaphors are then juxtaposed with a summary of the essay’s main topic: “intertwine creativity with my identity.” 

This essay’s main areas for improvement are grammatical. What Not to Wear should be italicized, “self-expression” should be hyphenated, and the last sentence could use the following tweaks to make it less of a run-on: “I hope that my future will open new doors for me, closet doors included, at the University of California. There, I will have opportunities to intertwine creativity with my identity even further.”

Since identity is the main topic of this essay, it would also be fitting for the writer to go into more depth about it. The immediate takeaways from the essay are that the writer is Asian and interested in fashion — however, more descriptions could be added to these parts. For example, the writer could replace Asian with Laotian-American and change a sentence in the second to last paragraph to “dressing up in everything from bell bottom jeans to oversized flannel shirts soon became what I looked forward to each morning.”

Prompt: What would you say is your greatest talent or skill? How have you developed and demonstrated that talent over time? (350 words)

Let’s fast-forward time. Strides were made toward racial equality. Healthcare is accessible to all; however, one issue remains. Our aquatic ecosystems are parched with dead coral from ocean acidification. Climate change has prevailed.

Rewind to the present day.

My activism skills are how I express my concerns for the environment. Whether I play on sandy beaches or rest under forest treetops, nature offers me an escape from the haste of the world. When my body is met by trash in the ocean or my nose is met by harmful pollutants, Earth’s pain becomes my own. 

Substituting coffee grinds as fertilizer, using bamboo straws, starting my sustainable garden, my individual actions needed to reach a larger scale. I often found performative activism to be ineffective when communicating climate concerns. My days of reposting awareness graphics on social media never filled the ambition I had left to put my activism skills to greater use. I decided to share my ecocentric worldview with a coalition of environmentalists and host a climate change rally outside my high school.

Meetings were scheduled where I informed students about the unseen impact they have on the oceans and local habitual communities. My fingers were cramped from all the constant typing and investigating of micro causes of the Pacific Waste Patch, creating reusable flyers, displaying steps people could take from home in reducing their carbon footprint. I aided my fellow environmentalists in translating these flyers into other languages, repeating this process hourly, for five days, up until rally day. 

It was 7:00 AM. The faces of 100 students were shouting, “The climate is changing, why can’t we?” I proudly walked on the dewy grass, grabbing the microphone, repeating those same words. The rally not only taught me efficient methods of communication but it echoed my environmental activism to the masses. The City of Corona would be the first of many cities to see my activism, as more rallies were planned for various parts of SoCal. My once unfulfilled ambition was fueled by my tangible activism, understanding that it takes more than one person to make an environmental impact.

One of the largest strengths of this response is its speed. From the very beginning, we are invited to “fast-forward” and “rewind” with the writer. Then, after we focus ourselves in the present, this writer keeps their quick pace with sentences like “Substituting coffee grounds as fertilizer, using bamboo straws, starting my sustainable garden, my individual actions needed to reach a larger scale.” A common essay-writing blunder is using a predictable structure that loses the attention of the reader, but this unique pacing keeps things interesting.

Another positive of this essay is how their passion for environmental activism shines through. The essay begins by describing the student’s connection to nature (“nature offers me an escape from the haste of the world”), moves into discussing the personal actions they have taken (“substituting coffee grounds as fertilizer”), and then explains the rally the student hosted. While the talent the student is writing about is their ability to inspire others to fight against climate change, establishing the personal affinity towards nature and individual steps they took demonstrate the development of their passion. This makes their talent appear much more significant and unique. 

This essay could be improved by being more specific about what this student’s talent is. There is no sentence that directly states what this student considers to be their talent. Although the essay is still successful at displaying the student’s personality, interests, and ambition, by not explicitly mentioning their talent, they leave it up to the reader’s interpretation.

Depending on how quickly they read the essay or how focused they are, there’s a possibility the reader will miss the key talent the student wanted to convey. Making sure to avoid spoon-feeding the answer to their audience, the student should include a short sentence that lays out what they view as their main talent.

At six, Mama reads me a story for the first time. I listen right up until Peter Pan talks about the stars in the night sky. “What’s the point of stars if they can’t be part of something?” Mama looks at me strangely before closing the book. “Sometimes, looking on is more helpful than actively taking part. Besides, stars listen- like you. You’re a good listener, aren’t you?” I nod. At eleven, my sister confides in me for the first time. She’s always been different, in a way even those ‘mind doctors’ could never understand. I don’t understand either, but I do know that I like my sister. She’s mean to me, but not like people are to her. She tells me how she sees the world, and chokes over her words in a struggle to speak. She trusts me, and that makes me happy. So, I listen. I don’t speak; this isn’t a story where I speak. At sixteen, I find myself involved with an organization that provides education to rural children. Dakshata is the first person I’ve tutored in Hindi. She’s also my favorite. So, when she interrupts me mid-lesson one evening, lips trembling and eyes filling with tears, I decide to put my pen down and listen. I don’t speak; I don’t take part in this story. Later, as I hug the girl, I tell her about the stars and how her mother is among their kind- unable to speak yet forever willing to listen. Dakshata now loves the stars as much as I do. At seventeen, I realize that the first thing that comes to my mind when someone asks me about a skill I possess is my ability to listen. Many don’t see it as a skill, and I wouldn’t ask them to either, but it’s important. When you listen, you see, you need not necessarily understand, but you do comprehend. You empathize on a near-cosmic level with the people around you and learn so much more than you ever thought possible. Everything is a part of something- even the stars with their ears.

The essay as a whole is an excellent example of narrative-based writing. The narrative begins with a captivating hook. The first sentence catches the reader by surprise, since it does not directly respond to the prompt by naming the writer’s greatest talent or skill. Instead, it tells a childhood story which does not seem to be related to a skill at first. This creates intrigue, and the second sentence adds to it by introducing a conflict. It causes readers to wonder why Peter Pan’s stargazing would make a six year old stop listening — hooked into the story, they continue reading.

The writer continues to create a moving narrative by using dialogue. Dialogue allows the writer to show rather than tell , which is a highly effective way to make an essay convey emotion and keep readers’ attention. The writer also shows their story by using language such as “mind doctors” instead of “psychologists” — this immerses readers in the author’s perspective as an 11 year old at the time. 

Two motifs, or recurring themes, tie the essay together: listening and looking at the stars. The last paragraph powerfully concludes the essay by explaining these themes and circling back to the introduction.

Crafting transitions is one area where this essay could be improved. The paragraph after “I nod” begins abruptly, and without any sentence to connect the writer’s dialogue at age six with her experiences at age 11. One way to make the transition smoother would be to begin the paragraph after “I nod” with “I try to be a good listener again at eleven, when my sister confides in me for the first time.”

This essay would also be more impactful if the writer explained what they aspire to do with their ability to listen in the future. While it is most important for your essay to explain how your past experiences have made you who you are in the present, looking towards the future allows admissions readers to imagine the impact you might make after graduation. The writer could do this in the last paragraph of their essay by writing the following: “Many don’t see it as a skill, and I wouldn’t ask them to either, but I find it important — especially as an aspiring social worker.”

Prompt: Think about an academic subject that inspires you. Describe how you have furthered this interest inside and/or outside of the classroom. (350 words)

I distinctly remember the smile on Perela’s face when she found out her mother would be nursed back to health. I first met Perela and her mother at the Lestonnac Free Clinic in San Bernardino where I volunteered as a Spanish translator. I was in awe of the deep understanding of biology that the medical team employed to discover solutions. Despite having no medical qualifications of my own, I realized that by exercising my abilities to communicate and empathize, I could serve as a source of comfort and encouragement for Perela and her mother. The opportunity to combine my scientific curiosity and passion for caring for people cultivated my interest in a career as a physician.

To further explore this interest, I attended a summer medical program at Georgetown University. I participated in lectures on circulation through the heart, practiced stitches on a chicken leg, and assisted in giving CPR to a dummy in the patient simulation laboratory. Every fact about the human body I learned brought with it ten new questions for me to research. I consistently stayed after each lecture to gain insight about how cells, tissues, and organs all work together to carry out immensely complicated functions. The next year, in my AP Biology class, I was further amazed with the interconnected biological systems as I learned about the relationships between the human body and ecosystems. I discussed with my teacher how environmental changes will impact human health and how we must broaden our perspectives to use medicine to tackle these issues.

By integrating environmental and medical science, we can develop effective solutions to reduce the adverse effects of environmental degradation that Perela’s mother may have faced unintentionally. I want to go into the medical field so I can employ a long-term approach to combat biology’s hidden anomalies with a holistic viewpoint. I look forward to utilizing my undergraduate classes and extracurriculars to prepare for medical school so I can fight for both health care and environmental protection.

This student primarily answers the prompt in their middle paragraph as they describe their experience at a summer medical program as well as their science coursework in high school. This content shows their academic curiosity and rigor, yet the best part of the essay isn’t the student’s response to the prompt. The best part of this essay is the way the student positions their interest in medicine as authentic and unique.

The student appears authentic when they admit that they haven’t always been interested in medical school. Many applicants have wanted to be doctors their whole life, but this student is different. They were just in a medical office to translate and help, then got hooked on the profession and took that interest to the next level by signing up for a summer program.

Additionally, this student positions themself as unique as they describe the specifics of their interest in medicine, emphasizing their concern with the ways medicine and the environment interact. This is also refreshing!

Of course, you should always answer the prompt, but it’s important to remember that you can make room within most prompts to say what you want and show off unique aspects of yourself—just as this student did.

One thing this student should be careful of is namedropping Georgetown for the sake of it. There is no problem in discussing a summer program they attended that furthered their interest in medicine, but there is a problem when the experience is used to build prestige. Admissions officers already know that this student attended a summer program at Georgetown because it’s on their application. The purpose of the essay is to show  why attending the program was a formative moment in their interest.

The essay gets at the  why a bit when it discusses staying after class to learn more about specific topics, but the student could have gone further in depth. Rather than explaining the things the student did during the program, like stitching chicken legs and practicing CPR, they should have continued the emotional reflection from the first paragraph by describing what they thought and felt when they got hands-on medical experience during the program. 

Save describing prestigious accomplishments for your extracurriculars and resume; your essay is meant to demonstrate what made you you.

I love spreadsheets.

It’s weird, I know. But there’s something endlessly fascinating about taking a bunch of raw numbers, whipping and whacking them into different shapes and forms with formulas and equations to reveal hidden truths about the universe. The way I like to think about it is that the universe has an innate burning desire to tell us its stories. The only issue is its inability to talk with us directly. Most human stories are written in simple words and letters, but the tales of the universe are encrypted in numbers and relationships, which require greater effort to decode to even achieve basic comprehension. After all, it took Newton countless experimentation to discover the love story between mass and gravitation.

In middle school, whenever I opened a spreadsheet, I felt like I was part of this big journey towards understanding the universe. It took me a couple of years, but I eventually found out that my interest had a name: Data Science. With this knowledge, I began to read extensively about the field and took online courses in my spare time. I found out that the spreadsheets I had been using was just the tip of the iceberg. As I gained more experience, I started using more powerful tools like R (a statistical programming language) which allowed me to use sophisticated methods like linear regressions and decision trees. It opened my eyes to new ways to understand reality and changed the way I approached the world.

The thing I love most about data science is its versatility. It doesn’t matter if the data at hand is about the airflow on an owl’s wing or the living conditions of communities most crippled by poverty. I am able to utilize data science to dissect and analyze issues in any field. Each new method of analysis yields different stories, with distinct actors, settings, and plots. I’m an avid reader of the stories of the universe, and one day I will help the world by letting the universe write its own narrative.

This is an essay that draws the reader in. The student’s candid nature and openness truly allows us to understand why they are fascinated with spreadsheets themself, which in turn makes the reader appreciate the meaning of this interest in the student’s life. 

First, the student engages readers with their conversational tone, beginning “I love spreadsheets. It’s weird, I know,” followed shortly after by the phrase “whipping and whacking.” Then, they introduce their idea to us, explaining how the universe is trying to tell us something through numbers and saying that Newton discovered “the love story between mass and gravitation,” and we find ourselves clearly following along. They put us right there with them, on their team, also trying to discover the secrets of the universe. It is this bond between the student and the reader that makes the essay so engaging and worth reading.

Because the essay is focused on the big picture, the reader gets a sense of the wide-eyed wonderment this student experiences when they handle and analyze data. The student takes us on the “big journey towards understanding the universe” through the lens of Data Science. Explaining both the tools the student has used, like R and statistical regression, and the ideas the student has explored, like owl’s wings and poverty, demonstrates how this student fits into the micro and macro levels of Data Science. The reader gets a complete picture of how this student could change the world through this essay—something admissions officers always want to see.

The biggest thing that would improve this essay is an anecdote. As it’s written, the essay looks at Data Science from a more theoretical or aspirational perspective. The student explains all that Data Science can enable, but besides for explaining that they started coding with spreadsheets and R, they provide very little personal experience working with Data Science. This is where an anecdote would elevate the essay.

Adding a story about the first data set they examined or an independent project they undertook as a hobby would have elicited more emotion and allowed for the student to showcase their accomplishments and way of thinking. For example, they could delve into the feeling of enlightenment that came from first discovering a pattern in the universe. Or maybe they could describe how analyzing data was the catalyst that led them to reach out to local businesses to help them improve their revenue. 

If you have an impactful and enduring interest, such as this student does, you will have at least one anecdote you could include in your essay. You’ll find that essays with anecdotes are able to work in more emotional reflection that make the essay more memorable and the student more likable.

Prompt: What have you done to make your community a better place? (350 words)

Blinking sweat from my eyes, I raised my chin up to the pullup bar one last time before dropping down, my muscles trembling. But despite my physical exhaustion at the end of the workout, mentally, I felt reinvigorated and stronger than ever.

Minutes later, I sat at my computer, chatting with my friends about our first week in quarantine. After listening to numerous stories concerning boredom and loneliness, it struck me that I could use my passion for fitness to help my friends—I jumped at the chance to do so. 

After scouring the internet for the most effective exercises and fitness techniques, I began hosting Zoom workouts, leading friends, family, and anyone else who wanted to join in several fun exercises each week. I hoped these meetings would uplift anyone struggling during quarantine, whether from loneliness, uncertainty, or loss of routine. I created weekly workout plans, integrating cardio, strength, and flexibility exercises into each. Using what I learned from skating, I incorporated off-ice training exercises into the plans and added stretching routines to each session. 

Although many members were worried that they wouldn’t be able to complete exercises as well as others and hesitated to turn their cameras on, I encouraged them to show themselves on screen, knowing we’d only support one another. After all, the “face-to-face” interactions we had while exercising were what distinguished our workouts from others online; and I hoped that they would lead us to grow closer as a community. 

As we progressed, I saw a new-found eagerness in members to show themselves on camera, enjoying the support of others. Seeing how far we had all come was immensely inspiring: I watched people who couldn’t make it through one circuit finish a whole workout and ask for more; instead of staying silent during meetings, they continually asked for tips and corrections.

Despite the limitations placed on our interactions by computer screens, we found comfort in our collective efforts, the camaraderie between us growing with every workout. For me, it confirmed the strength we find in community and the importance of helping one another through tough times.

This essay accomplishes three main goals: it tells a story of how this student took initiative, it explores the student’s values, and it demonstrates their emotional maturity. We really get a sense of how this student improved their community while also gaining a large amount of insight into what type of person this student is.

With regards to initiative, this student writes about a need they saw in their community and the steps they took to satisfy that need. They describe the extensive thought that went into their decisions as they outline the planning of their classes and their unique decision to incorporate skating techniques in at-home workouts.

Additionally, they explore their values, including human connection. The importance of connection to this student is obvious throughout the essay as they write about their desire “to grow closer as a community.” It is particularly apparent with their final summarizing sentence: “For me, it confirmed the strength we find in community and the importance of helping one another through tough times.”

Lastly, this student positions themself as thoughtful when they recognize the way that embarrassment can get in the way of forming community. They do this through the specific example of feeling embarrassment when turning on one’s camera during a video call—a commonly-felt feeling. This ability to recognize fear of embarrassment as an obstacle to camaraderie shows maturity on the part of this applicant. 

This essay already has really descriptive content, a strong story, and a complete answer to the prompt, however there is room for every essay to improve. In this case, the student could have worked more descriptive word choice and figurative language into their essay to make it more engaging and impressive. You want your college essay to showcase your writing abilities as best as possible, while still sounding like you.

One literary device that would have been useful in this essay is a conceit or an extended metaphor . Essays that utilize conceits tend to begin with a metaphor, allude to the metaphor during the body of the paragraph, and end by circling back to the original metaphor. All together, it makes for a cohesive essay that is easy to follow and gives the reader a satisfying opening and conclusion to the essay.

The idea at the heart of this essay—working out to strengthen a community—would make for a great conceit. By changing the anecdote at the beginning to maybe reflect the lack of strength the student felt when working out alone and sprinkling in words and phrases that allude to strength and exercise during the essay, the last sentence (“For me, it confirmed the strength we find in community and the importance of helping one another through tough times”) would feel like a fulfilling end to the conceit rather than just a clever metaphor thrown in. 

Prompt: What have you done to make your school or your community a better place? (350 words)

The scent of eucalyptus caressed my nose in a gentle breeze. Spring had arrived. Senior class activities were here. As a sophomore, I noticed a difference between athletic and academic seniors at my high school; one received recognition while the other received silence. I wanted to create an event celebrating students academically-committed to four-years, community colleges, trades schools, and military programs. This event was Academic Signing Day.

The leadership label, “Events Coordinator,” felt heavy on my introverted mind. I usually was setting up for rallies and spirit weeks, being overlooked around the exuberant nature of my peers. 

I knew a change of mind was needed; I designed flyers, painted posters, presented powerpoints, created student-led committees, and practiced countless hours for my introductory speech. Each committee would play a vital role on event day: one dedicated to refreshments, another to technology, and one for decorations. The fourth-month planning was a laborious joy, but I was still fearful of being in the spotlight. Being acknowledged by hundreds of people was new to me. 

The day was here. Parents filled the stands of the multi-purpose room. The atmosphere was tense; I could feel the angst building in my throat, worried about the impression I would leave. Applause followed each of the 400 students as they walked to their college table, indicating my time to speak. 

I walked up to the stand, hands clammy, expression tranquil, my words echoing to the audience. I thought my speech would be met by the sounds of crickets; instead, smiles lit up the stands, realizing my voice shone through my actions. I was finally coming out of my shell. The floor was met by confetti as I was met by the sincerity of staff, students, and parents, solidifying the event for years to come. 

Academic students were no longer overshadowed. Their accomplishments were equally recognized to their athletic counterparts. The school culture of athletics over academics was no longer imbalanced. Now, everytime I smell eucalyptus, it is a friendly reminder that on Academic Signing Day, not only were academic students in the spotlight but so was my voice.

This is a good essay because it describes the contribution the student made to their community and the impact that experience had on shaping their personality. Admissions officers get to see what this student is capable of and how they have grown, which is important to demonstrate in your essays. Throughout the essay there is a nice balance between focusing on planning the event and the emotions it elicited from this student, which is summed up in the last sentence: “not only were academic students in the spotlight but so was my voice.”

With prompts like this one (which is essentially a Community Service Essay ) students sometimes take very small contributions to their community and stretch them—oftentimes in a very obvious way. Here, the reader can see the importance of Academic Signing Day to the community and the student, making it feel like a genuine and enjoyable experience for all involved. Including details like the four months of planning the student oversaw, the specific committees they delegated tasks to, and the hundreds of students and parents that attended highlights the skills this student possesses to plan and execute such a large event.

Another positive aspect of this essay is how the student’s emotions are intertwined throughout the essay. We see this student go from being a shy figure in the background to the confident architect of a celebrated community event, all due to their motivation to create Academic Signing Day. The student consistently shows throughout the essay, instead of telling us what happened. One example is when they convey their trepidation to public speaking in this sentence: “I walked up to the stand, hands clammy, expression tranquil, my words echoing to the audience. I thought my speech would be met by the sounds of crickets.”

Employing detailed descriptions of feelings, emotions, fears, and body language all contribute to an essay that reveals so much in subtle ways. Without having to be explicitly told, the reader learns the student is ambitious, organized, a leader, and someone who deeply values academic recognition when they read this essay.

While this essay has many positives, there are a couple of things the student could work on. The first is to pay more attention to grammar. There was one obvious typo where the student wrote “the fourth-month planning was a laborious joy”, but there were also many sentences that felt clunky and disjointed. Each and every essay you submit should put your best foot forward and impress admissions officers with your writing ability, but typos immediately diminish your credibility as a writer and sincerity as an applicant.

It’s important to read through your essay multiple times and consider your specific word choice—does each word serve a purpose, could a sentence be rewritten to be less wordy, etc? However, it’s also important you have at least one other person edit your essay. Had this student given their essay to a fresh set of eyes they might have caught the typo and other areas in need of improvement.

Additionally, this student began and ended the essay with the smell of eucalyptus. Although this makes for an intriguing hook, it has absolutely nothing to do with the actual point of the essay. It’s great to start your essay with an evocative anecdote or figurative language, but it needs to relate to your topic. Rather than wasting words on eucalyptus, a much stronger hook could have been the student nervously walking up to the stage with clammy hands and a lump in their throat. Beginning the essay with a descriptive sentence that puts us directly into the story with the student would draw the reader in and get them excited about the topic at hand.

Prompt: What have you done to make your school or community a better place? (350 words) 

“I wish my parents understood.” Sitting at the lunch table, I listened as my friends aired out every detail of their life that they were too afraid to share with their parents. Sexuality, relationships, dreams; the options were limitless. While I enjoyed playing therapist every 7th period, a nagging sensation that perhaps their parents should understand manifested in me. Yet, my proposal was always met with rolling eyes; “I wish they understood” began every conversation, but nothing was being done beyond wishing on both sides. 

I wanted to help not just my friends but the countless other stories I was told of severed relationships and hidden secrets. Ultimately, my quest for change led me to BFB, a local nonprofit. Participating in their Youth Leadership program, I devised and implemented a plan for opening up the conversation between students and parents with the team I led. We successfully hosted relationship seminars with guest speakers specializing on a range of topics, from inclusive education to parental pressure, and were invited to speak for BFB at various external events with local government by the end of my junior year. Collaborating with mental health organizations and receiving over $1,000 in funding from international companies facilitated our message to spread throughout the community and eventually awarded us with an opportunity to tackle a research project studying mental health among teens during the pandemic with professors from the University at Buffalo and UC Los Angeles. 

While these endeavors collectively facilitated my team to win the competition, the most rewarding part of it all was receiving positive feedback from my community and close friends. “I wish my parents understood” morphed into “I’m glad they tried to understand”. I now lead a separate program under BFB inspired by my previous endeavors, advancing its message even further and leaving a legacy of change and initiative for future high schoolers in the program. As I leave for college, I hope to continue this work at the University of California and foster a diverse community that embraces understanding and growth across cultures and generations.

The essay begins with a strong, human-centered story that paints a picture of what the writer’s community looks like. The first sentence acts as a hook by leaving readers with questions — whose parents are being discussed, and what don’t they understand? With their curiosity now piqued, readers become intrigued enough to move on to the next sentences. The last sentence of the first paragraph and beginning of the second relate to the same topic of stories from friends, making for a highly effective transition.

The writer then does a great job of describing their community impact in specific detail, which is crucial for this prompt. Rather than using vague and overly generalized language, the writer highlights their role in BFB with strong action verbs like “devised” and “implemented.” They also communicate the full scope of their impact with quantifiable metrics like “$1,000 in funding,” all while maintaining a flowing narrative style.

The essay ends by circling back to the reason why the writer got involved in improving their community through BFB, which makes the essay more cohesive and moving. The last sentences connect their current experiences improving community with their future aspirations to do so, both in the wider world and at a UC school. This forward-looking part allows admissions officers to get a sense of what the writer might accomplish as a UC alum/alumna, and is certainly something to emulate.

This essay’s biggest weakness is its organization. Since the second paragraph contains lots of dense information about the writer’s role in BFB, it would benefit from a few sentences that tie it back to the narrative in the first paragraph. For instance, the third sentence of the paragraph could be changed like so: “Participating in their Youth Leadership program, I led my team through devising and implementing a plan to foster student-parent conversations — the ones that my 7th period friends were in need of.”

The last paragraph also has the potential to be reorganized. The sentence with the “I wish my parents understood” quote would be more powerful at the end of the paragraph rather than in the middle. With a short transition added to the beginning, the new conclusion would look like so: “ Through it all, I hope to help ‘I wish my parents understood’ morph into ‘I’m glad they tried to understand’ for my 7th period friends and many more.” 

I drop my toothbrush in the sink as I hear a scream. Rushing outside, I find my mom’s hand painfully wedged in the gap between our outward-opening veranda doors. I quickly open it, freeing her hand as she gasps in relief. 

As she ices her hand, I regard the door like I would a trivia question or math problem – getting to know the facts before I start working on a solution. I find that, surprisingly, there is not a single protrusion to open the door from the outside! 

Perhaps it was the fact that my mom couldn’t drive or that my dad worked long hours, but the crafts store was off-limits; I’ve always ended up having to get resourceful and creative with whatever materials happened to be on hand in order to complete my impromptu STEM projects or garage builds. Used plastic bottles of various shapes and sizes became buildings for a model of a futuristic city. Cylindrical capacitors from an old computer, a few inches in height, became scale-size storage tanks. 

Inspired by these inventive work-arounds and spurred on by my mom’s plight, I procure a Command Strip, a roll of tennis racket grip, and, of course, duct tape. I fashion a rudimentary but effective solution: a pull handle, ensuring she would never find herself stuck again.

A desire to instill others in my community with this same sense of resourcefulness led me to co-found “Repair Workshops” at my school – sessions where we teach students to fix broken objects rather than disposing of them. My hope is that participants will walk away with a renewed sense of purpose to identify problems faced by members of their community (whether that’s their neighbor next door or the planet as a whole) and apply their newfound engineering skills towards solutions.

As I look towards a degree and career in engineering and business, these connections will serve as my grounding point: my reminder that in disciplines growing increasingly quantitative, sometimes the best startup ideas or engineering solutions originate from a desire to to better the lives of people around me.

This essay is a good example of telling a story with an authentic voice. With its down-to-earth tone and short, punchy paragraphs, it stands out as a piece of writing that only the author could have written. That is an effective way for you to write any of your college essays as well.

After readers are hooked by the mention of screaming in the first sentence, the writer immerses the readers in their thinking. This makes the essay flow very naturally — rather than a first paragraph of narrative followed by an unrelated description of STEM projects, the whole essay is a cohesive story that shows how the writer came to improve their community. 

Their take on community also makes the essay stand out. While many responses to this prompt will focus on an amorphous, big-picture concept of community, such as school or humanity, this essay is about a community that the writer has a close connection to — their family. Family is also not the large group of people that most applicants would first attach to the word “community,” but writing about it here is a creative take on the prompt. Though explaining community impact is most important, choosing the most unique community you are a part of is a great way to make your essay stand out.

This essay’s main weakness is that the paragraph about Repair Workshops does not go into enough detail about community impact. The writer should highlight more specific examples of leadership here, since it would allow them to demonstrate how they hope to impact many more communities besides their family. 

After the sentence ending with “fix broken objects rather than disposing of them,” a new part could be added that shows how the writer taught students. For example, the writer could tell the story of how “tin cans became compost bins” as they explained the importance of making the world a better place. 

Then, at the end of the paragraph, the writer could more concretely explain the visions they have to expand the impact of Repair Workshops. A good concluding sentence could start with “I too hope to use engineering skills and resourcefulness to…” Adding this extra context would also make the paragraph transition better to the final paragraph of the essay, which somewhat abruptly begins by mentioning the writer’s previously unmentioned career interests in engineering and business.

Where to Get Feedback on Your UC Essays

Want feedback like this on your University of California essays before you submit? We offer expert essay review by advisors who have helped students get into their dream schools. You can book a review with an expert to receive notes on your topic, grammar, and essay structure to make your essay stand out to admissions officers. In fact, Alexander Oddo , an essay expert on CollegeVine, provided commentary on several of the essays in this post.

Haven’t started writing your essay yet? Advisors on CollegeVine also offer expert college counseling packages . You can purchase a package to get one-on-one guidance on any aspect of the college application process, including brainstorming and writing essays.

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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.

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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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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. 

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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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Deepansh Gautam

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Guest Essay

Why Biden Has a Narrower Path to the Presidency Than Trump, in 11 Maps

Illustration by Akshita Chandra/The New York Times; Images by PhotoObjects.net, Yuji Sakai, and THEPALMER/Getty Images

By Doug Sosnik Graphics by Quoctrung Bui

Mr. Sosnik was a senior adviser to President Bill Clinton from 1994 to 2000 and has advised over 50 governors and U.S. senators.

While polls show the race for president is tightening, Joe Biden still has a narrower and more challenging path to winning the election than Donald Trump. The reason is the Electoral College: My analysis of voter history and polling shows a map that currently favors Mr. Trump, even though recent developments in Arizona improve Mr. Biden’s chances. The Biden campaign will need to decide this summer which states to contest hardest. Our Electoral College maps below lay out the best scenarios for him and Mr. Trump.

Seven states with close results determined who won both the 2020 and the 2016 presidential elections, and those same seven states will most likely play the same battleground role this fall: three industrial states – Michigan, Pennsylvania and Wisconsin – and four Sun Belt states – Arizona, Georgia, Nevada and North Carolina.

The seven states that will most likely decide the 2024 presidential election

Mr. Biden’s declining popularity in the Sun Belt states is the main reason Mr. Trump has an edge right now. He is especially struggling with young and nonwhite voters there. Let’s take a closer look:

According to 2020 exit polls , Mr. Biden won 65 percent of Latino voters, who comprised roughly a fifth of voters in Arizona and Nevada. And Mr. Biden won 87 percent of Black voters, who made up 29 percent of the Georgia vote and 23 percent of the North Carolina vote. He also won 60 percent of voters aged 18 to 29. Now look at this year: A New York Times/Siena College poll released last weekend showed support for Mr. Biden had dropped 18 points with Black voters, 15 points with Latinos and 14 points with younger voters nationally.

Abortion could be a decisive issue in Mr. Biden stemming this erosion of support in Arizona and Nevada. The Arizona Supreme Court’s ruling last week that largely bans abortions raises the stakes of a likely ballot initiative on the issue there in November. It also appears likely that there will be a similar ballot measure in Nevada.

Nevertheless, the key to Mr. Biden’s victory is to perform well in the three industrial states. If Mr. Trump is able to win one or more of Pennsylvania, Michigan and Wisconsin, Mr. Biden’s path to 270 electoral votes becomes even narrower.

If Mr. Biden and Mr. Trump remain ahead in the states where they are currently running strongest, the outcome of the election could come down to who wins Michigan and the two Sun Belt states where abortion will very likely be on the ballot, Arizona and Nevada.

Based on past voting, Mr. Trump will start out the general election with 219 electoral votes, compared to 226 votes for Mr. Biden, with 93 votes up for grabs.

Voter history and recent polling suggest that Mr. Trump is in a strong position to win North Carolina . Republicans have carried the state in every presidential election since 1976 except in 2008. In a Wall Street Journal battleground poll taken in March, Mr. Biden had only 37 percent job approval in the state. By winning North Carolina , Mr. Trump would have 235 electoral votes and two strong paths to 270.

The first path involves carrying Georgia , a state he lost by fewer than 12,000 votes in 2020. Before then, Republicans won Georgia in every election since 1992. If Mr. Trump carries North Carolina and Georgia , he would have a base of 251 electoral votes with four scenarios that get him to 270.

Scenario 1 Then all Mr. Trump needs is Pennsylvania …

Scenario 2 … or Michigan and Nevada …

Scenario 3 … or Michigan and Arizona …

Scenario 4 … or Arizona and Wisconsin.

The second and harder path for Mr. Trump would be if he carried only one Southern swing state – most likely North Carolina . He would have only 235 electoral votes and would need to win three of the six remaining battleground states.

Scenario 5 Then he would need to win Arizona , Michigan and Wisconsin …

Scenario 6 … or Arizona , Nevada and Pennsylvania .

How Biden Can Win

It is difficult to see how Mr. Biden gets re-elected without doing well in the industrial battleground states – the so-called “ Blue Wall ” for Democrats. This is particularly true of Pennsylvania, given the state’s 19 electoral votes and Mr. Biden’s ties there and appeal to middle-class and blue-collar voters. That’s why he’s spending three days in Pennsylvania this week.

Mr. Biden will most likely need to win at least one other industrial battleground – with Wisconsin the most probable, since his polling numbers there are stronger than in the other battleground states.

A combination of factors have made winning Michigan much more challenging for Mr. Biden. Hamas’s attack on Israel and the war in Gaza have ripped apart the coalitions that enabled Democrats to do so well in the state since 2018. There are over 300,000 Arab Americans there, as well as a large Jewish population. Both groups were crucial to Mr. Biden’s success there in 2020.

In addition, Michigan voters’ perception of the economy is more negative compared with the other battleground states. In the Journal battleground poll , two-thirds of Michigan voters described the national economy negatively; more than half had a negative opinion of the state’s economy.

Now let’s look at Mr. Biden’s map.

Mr. Biden’s best strategy is based on winning Pennsylvania and Wisconsin, which would give him 255 electoral votes (assuming that he carries the 2nd Congressional District in Nebraska). By carrying these states, Mr. Biden has several paths to 270, but the first three scenarios are his most viable.

Scenario 1 He just needs to win Michigan …

Scenario 2 … or Arizona and Nevada …

Scenario 3 ... or Georgia .

There are two other scenarios where Mr. Biden loses Wisconsin and keeps Pennsylvania . But that would mean winning states where Mr. Biden is polling much worse.

Scenario 4 They involve Mr. Biden winning Georgia and Arizona …

Scenario 5 … or Michigan and Georgia .

A Look Ahead

With over six months to go until Election Day, given the volatility in the world and the weaknesses of Mr. Biden and Mr. Trump, it would be foolish to make firm predictions about specific results. And other electoral map scenarios are possible: Recent polling shows Mr. Biden with a narrow lead in Minnesota, a state that usually votes for Democrats for president. While it is mathematically possible for Mr. Biden to win without carrying Minnesota, it is unlikely he will be elected if he cannot carry this traditionally Democratic state.

For the third election cycle in a row, a small number of voters in a handful of states could determine the next president of the United States.

If the election remains close but Mr. Biden is unable to regain support from the core group of voters who propelled him to victory in 2020 — young and nonwhite voters — then we could be headed to a repeat of the 2016 election. The outcome of that election was decided by fewer than 80,000 votes in Michigan, Pennsylvania and Wisconsin.

Last week’s abortion ruling in Arizona, and the likely abortion ballot initiatives in that state and Nevada, give Mr. Biden the possibility of being re-elected even if he loses Michigan. That’s why, if we have another close presidential election, I think Arizona, Michigan and Nevada will likely determine the outcome for Mr. Biden or Mr. Trump.

Based on my experience as Bill Clinton’s White House political director in his 1996 re-election campaign, I would take immediate advantage of Mr. Biden’s significant fund-raising advantage over Mr. Trump to focus on shoring up the president’s chances in Michigan and the must-win states of Pennsylvania and Wisconsin, while at the same time trying to keep Georgia and North Carolina in play. Mr. Biden does not need to win either of those Sun Belt states to get re-elected, but draining Mr. Trump’s resources there could help him in other battleground states.

More on the 2024 presidential election

Democrats Need to Stop Playing Nice

Too often, Democrats litigate; Republicans fight.

By Joe Klein

One Purple State Is ‘Testing the Outer Limits of MAGAism’

North Carolina Republicans are “in the running for the most MAGA party in the nation.”

By Thomas B. Edsall

2024, Meet 1892, Your Doppelgänger

Great political change can unfold when the political system seems woefully stalled.

By Jon Grinspan

Doug Sosnik was a senior adviser to President Bill Clinton from 1994 to 2000 and has advised over 50 governors and U.S. senators.

The Times is committed to publishing a diversity of letters to the editor. We’d like to hear what you think about this or any of our articles. Here are some tips . And here’s our email: [email protected] .

Follow the New York Times Opinion section on Facebook , Instagram , TikTok , WhatsApp , X and Threads .