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Stanford Common App Essay: A meaningful background, identity, interest, or talent.

Some students have a background, identity, interest, or talent that is so meaningful they believe their application would be incomplete without it. if this sounds like you, then please share your story..

Once the lyrics started, my eyes had no chance of holding back the eager river of tears. B.o.B’s song “Don’t Let Me Fall” spoke to me deeply and immediately. It was the same morning I left the 2013 Mu Alpha Theta National Convention (the largest nationwide math competition), and upon the suggestion of my best friend Andrew, I listened to what he claimed would summarize our experience in a song. Every friend I made, every trophy I won, every game I played, all coalesced in my mind as I reminisced about what I instantly realized was the most jubilant week of my life. Never before had I experienced such a sharp pang of nostalgia, feeling just as blissful as I did melancholy.

Math chose me. Around the age of nine, I grasped the concept of special right triangles while staring at shower tiles. To this day, I habitually multiply numbers in my head, even when I sprint in a track meet or write an English essay. Independent of my math instinct, I possess a healthy competitive energy. Whether I dash down the Ultimate Frisbee field or race through Mario Kart courses, winning becomes more important than breathing. For me, the math competition is the perfect game. It satisfies my craving for challenge while encompassing my most natural instinct.

As intense as my zeal for math was, a more vibrant humanizing aspect slowly seeped into the experience. I noticed a steady change in how I spent my time at math competitions. I spent less time rechecking problems after the tests ended, and more time making sure I got to meet up with every friend I would otherwise never see. Math never decreased in importance, but became gradually overshadowed by the growing need to organize pick-up basketball and football games with other students in my free time. At the National Convention, I can always count on finding Hawaii’s Ka’iulani gossiping about her YouTube obsession, Tallahassee’s Jessie exuding her unmatched appetite for sports talk, and Seattle’s Tim enjoying crazy card games until the brink of curfew, all under one roof. Undoubtedly, many people enjoy the company of friends they seldom see. There are plenty I have not talked to in years, but here something different motivated me.

Maybe it was the camaraderie of spending time with people who inexplicably subjected their brains to more thinking on the weekends, or simply the fact that seeing each other was rare. Perhaps, though, our community most profoundly resonates from the unconscious thought that deep down, we shared some mathematical enlightenment. They, too, have seen the Pythagorean theorem’s relevance in the distance formula, and they, too, have comprehended the logic behind otherwise mindless area equations. Somehow, it felt as though we were all connected long before we met.

San Diego’s National Convention perfected my recipe for happiness: doubling the number of States I had friends in, threatening to overflow my trophy shelf (the family piano), and teaching me that math had far deeper implications than all the awards eventually only I would remember. My high school years have taught me to befriend my competitors, rather than alienate them. The tools and opportunities I have been blessed with, along with this lesson, have helped me build a confident identity for my adventures on the horizon.

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How mathematical practices can improve your writing

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

Caroline Yoon

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

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

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

Three mathematical practices that can improve your writing

Mathematical modelling.

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

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

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

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

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

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Problem-solving

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

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

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

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

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

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

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

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

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

Practical tips for productive writing beliefs and behaviours

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

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

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

If you would like advice and insight from academics and university staff delivered direct to your inbox each week,  sign up for the Campus newsletter .

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Course info, instructors.

• Prof. Haynes Miller
• Dr. Nat Stapleton
• Saul Glasman

Departments

• Mathematics

As Taught In

Learning resource types, project laboratory in mathematics.

Next: Revision and Feedback »

In this section, Prof. Haynes Miller and Susan Ruff describe the criteria for good mathematical writing and the components of the writing workshop .

A central goal of the course is to teach students how to write effective, journal-style mathematics papers. Papers are a key way in which mathematicians share research findings and learn about others’ work. For each research project, each student group writes and revises a paper in the style of a professional mathematics journal paper. These research projects are perfect for helping students to learn to write as mathematicians because the students write about the new mathematics that they discover. They own it, they are committed to it, and they put a lot of effort into writing well.

Criteria for Good Writing

In the course, we help students learn to write papers that communicate clearly, follow the conventions of mathematics papers, and are mathematically engaging.

Communicating clearly is challenging for students because doing so requires writing precisely and correctly as well as anticipating readers’ needs. Although students have read textbooks and watched lectures that are worded precisely, they are often unaware of the care with which each word or piece of notation was chosen. So when students must choose the words and notation themselves, the task can be surprisingly challenging. Writing precisely is even more challenging when students write about insights they’re still developing. Even students who do a good job of writing precisely may have a different difficulty: providing sufficient groundwork for readers. When students are deeply focused on the details of their research, it can be hard for them to imagine what the reading experience may be like for someone new to that research. We can help students to communicate clearly by pointing out places within the draft at which readers may be confused by imprecise wording or by missing context.

For most students, the conventions of mathematics papers are unfamiliar because they have not read—much less written—mathematics journal papers before. The students’ first drafts often build upon their knowledge of more familiar genres: humanities papers and mathematics textbooks and lecture notes. So the text is often more verbose or explanatory than a typical paper in a mathematics journal. To help students learn the conventions of journal papers, including appropriate concision, we provide samples and individualized feedback.

Finally, a common student preconception is that mathematical writing is dry and formal, so we encourage students to write in a way that is mathematically engaging. In Spring 2013, for example, one student had to be persuaded that he did not have to use the passive voice. In reality, effective mathematics writing should be efficient and correct, but it should also provide motivation, communicate intuition, and stimulate interest.

To summarize, instruction and feedback in the course address many different aspects of successful writing:

• Precision and correctness: e.g., mathematical terminology and notation should be used correctly.
• Audience awareness: e.g., ideas should be introduced with appropriate preparation and motivation.
• Genre conventions: e.g., in most mathematics papers, the paper’s conclusion is stated in the introduction rather than in a final section titled “Conclusion.”
• Style: e.g., writing should stimulate interest.
• Other aspects of effective writing, as needed.

To help students learn to write effective mathematics papers, we provide various resources, a writing workshop, and individualized feedback on drafts.

Writing Resources

Various resources are provided to help students learn effective mathematical writing.

The following prize-winning journal article was annotated to point out various conventions and strategies of mathematical writing. (Courtesy of Mathematical Association of America. Courtesy of a Creative Commons BY-NC-SA license.)

An Annotated Journal Article (PDF)

This document introduces the structure of a paper and provides a miscellany of common mistakes to avoid.

Notes on Writing Mathematics (PDF)

LaTeX Resources

The following PDF, TeX, and Beamer samples guide students to present their work using LaTeX, a high-quality typesetting system designed for the production of technical and scientific documentation. The content in the PDF and TeX documents highlights the structure of a generic student paper.

Sample PDF Document created by pdfLaTeX (PDF)

Sample TeX Document (TEX)

Beamer template (TEX)

The following resources are provided to help students learn and use LaTeX.

LaTeX-Project. “ Obtaining LaTeX .” August 28, 2009.

Downes, Michael. “Short Math Guide for LaTeX.” (PDF) American Mathematical Society . Version 1.09. March 22, 2002.

Oetiker, Tobias, Hubert Partl, et al. “The Not So Short Introduction to LaTeX 2ε.” (PDF) Version 5.01. April 06, 2011.

Reckdahl, Keith. “Using Imported Graphics in LaTeX and pdfLaTeX.” (PDF) Version 3.0.1. January 12, 2006.

Writing Workshop

Each semester there is a writing workshop, led by the lead instructor, which features examples to stimulate discussion about how to write well. In Spring 2013, Haynes ran this workshop during the third class session and used the following slide deck, which was developed by Prof. Paul Seidel and modified with the help of Prof. Tom Mrowka and Prof. Richard Stanley.

The 18.821 Project Report (PDF)

This workshop was held before students had begun to think about the writing component of the course, and it seemed as if the students had to be reminded of the lessons of the workshop when they actually wrote their papers. In future semesters, we plan to offer the writing workshop closer to the time that students are drafting their first paper. We may also focus the examples used in the workshop on the few most important points rather than a broad coverage.

• Download video

This video features the writing workshop from Spring 2013 and includes instruction from Haynes as well as excerpts of the class discussion.

« Previous: Writing | Next: Sample Student Papers »

In this section, Prof. Haynes Miller and Susan Ruff describe how students receive feedback on their writing and what is expected from students during the revision process.

Feedback and revision are critical to students’ development as mathematical writers in the course. For each project, each student team is required to write a first draft, meet with course instructors for a debriefing meeting, make revisions, and submit a final draft. This process provides an opportunity for a mid-project check-in about the students’ writing as well as their research, and it pushes them to produce a stronger final draft than what most could have managed on their own.

In the best situations, a team’s first draft represents the students’ best efforts but is still somewhat rough; we give them lots of feedback for reworking their paper, and their final draft is substantially clearer and more rigorous, well-motivated, and technically precise. In our experience, each subsequent paper is typically better than the one before.

Instructor Feedback on Writing

After a team submits its first draft, the team’s mentor for that project, and sometimes Haynes and sometimes Susan, reads the paper and crafts feedback. First drafts typically have plenty of room for improvement. We try not to overwhelm students with a huge number of comments; commenting on everything often leads to students getting lost in the details and unable to distinguish the most important points from more trivial points. Instead, we draw attention to the most important things for the students to improve. We try to craft constructive comments so that, rather than being discouraged, students will be inspired to revise. Sometimes a second round of revision is necessary. This whole process is quite like the refereeing process for journal articles.

Debriefing Meetings

Students receive feedback on their draft at a team debriefing meeting, which usually occurs several days after the first draft is submitted. Sharing feedback via the debriefing meeting provides two key advantages:

• Clarity and emphasis via discussion. Speaking face-to-face allows us to emphasize the most important feedback; to ask students questions and understand the intentions behind their writing; and to have some back-and-forth to make sure that students understand the feedback.
• Efficiency. Reading papers and commenting on papers takes a long time. The debriefings allow us to convey some of the feedback efficiently in person rather than on paper.

Most students take the debriefing sessions very seriously. They do not see our feedback on their work beforehand, and they are naturally curious and may be somewhat anxious, especially the first time. The face-to-face interaction always helps to frame suggestions in a constructive manner, and students almost never respond defensively. They generally listen attentively and make a sincere effort to respond to our critiques.

Immediately after the debriefings, we scan the marked-up papers and send an electronic copy to the team members. The final draft is typically due a week after the debriefing, giving students time to think about research extensions of their work and to improve their writing.

Self- and Peer-Editing

One of the things we look for in papers for the course is consistency of voice and notation among sections written by different team members. We encourage the students to help each other revise.

« Previous: Revision and Feedback

To illustrate the writing and revision process for the student papers, two sample projects are presented below.

Sample Paper 1: The Dynamics of Successive Differences Over ℤ and ℝ

This project developed from the project description for Number Squares (PDF) . To view the practice presentation and final presentation from this team of students, see the Sample Student Presentations page.

The student work is courtesy of Yida Gao, Matt Redmond, and Zach Steward. Used with permission.

• First Draft of Sample Paper 1 (PDF)
• First Draft of Sample Paper 1 with Comments from Susan Ruff (PDF - 2.5MB)
• First Draft of Sample Paper 1 with Comments from Prof. Haynes Miller (PDF - 3.6MB)
• Additional Comments on Sample Paper 1 from Prof. Haynes Miller (PDF)
• Final Version of Sample Paper 1 (PDF)

Debriefing for First Draft of Sample Paper 1

This video features the debriefing meeting for the first draft of Sample Paper 1. The student team first presents their findings, and then the course instructors offer feedback and discuss the mathematics and the writing for the project.

Sample Paper 2: Tossing a Coin

This project developed from the project description for Tossing a Coin (PDF) .

The student work is courtesy of Jean Manuel Nater, Peter Wear, and Michael Cohen. Used with permission.

• First Draft of Sample Paper 2 (PDF)
• First Draft of Sample Paper 2 with Comments from Susan Ruff (PDF - 1.7MB)
• First Draft of Sample Paper 2 with Comments from Prof. Haynes Miller (PDF - 2.7MB)
• Additional Comments on Sample Paper 2 from Prof. Haynes Miller (PDF)
• Final Version of Sample Paper 2 (PDF)

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53 Stellar College Essay Topics to Inspire You

College Essays

Most colleges and universities in the United States require applicants to submit at least one essay as part of their application. But trying to figure out what college essay topics you should choose is a tricky process. There are so many potential things you could write about!

In this guide, we go over the essential qualities that make for a great college essay topic and give you 50+ college essay topics you can use for your own statement . In addition, we provide you with helpful tips for turning your college essay topic into a stellar college essay.

What Qualities Make for a Good College Essay Topic?

Regardless of what you write about in your personal statement for college , there are key features that will always make for a stand-out college essay topic.

#1: It’s Specific

First off, good college essay topics are extremely specific : you should know all the pertinent facts that have to do with the topic and be able to see how the entire essay comes together.

Specificity is essential because it’ll not only make your essay stand out from other statements, but it'll also recreate the experience for admissions officers through its realism, detail, and raw power. You want to tell a story after all, and specificity is the way to do so. Nobody wants to read a vague, bland, or boring story — not even admissions officers!

For example, an OK topic would be your experience volunteering at a cat shelter over the summer. But a better, more specific college essay topic would be how you deeply connected with an elderly cat there named Marty, and how your bond with him made you realize that you want to work with animals in the future.

Remember that specificity in your topic is what will make your essay unique and memorable . It truly is the key to making a strong statement (pun intended)!

#2: It Shows Who You Are

In addition to being specific, good college essay topics reveal to admissions officers who you are: your passions and interests, what is important to you, your best (or possibly even worst) qualities, what drives you, and so on.

The personal statement is critical because it gives schools more insight into who you are as a person and not just who you are as a student in terms of grades and classes.

By coming up with a real, honest topic, you’ll leave an unforgettable mark on admissions officers.

#3: It’s Meaningful to You

The very best college essay topics are those that hold deep meaning to their writers and have truly influenced them in some significant way.

For instance, maybe you plan to write about the first time you played Skyrim to explain how this video game revealed to you the potentially limitless worlds you could create, thereby furthering your interest in game design.

Even if the topic seems trivial, it’s OK to use it — just as long as you can effectively go into detail about why this experience or idea had such an impact on you .

Don’t give in to the temptation to choose a topic that sounds impressive but doesn’t actually hold any deep meaning for you. Admissions officers will see right through this!

Similarly, don’t try to exaggerate some event or experience from your life if it’s not all that important to you or didn’t have a substantial influence on your sense of self.

#4: It’s Unique

College essay topics that are unique are also typically the most memorable, and if there’s anything you want to be during the college application process, it’s that! Admissions officers have to sift through thousands of applications, and the essay is one of the only parts that allows them to really get a sense of who you are and what you value in life.

If your essay is trite or boring, it won’t leave much of an impression , and your application will likely get immediately tossed to the side with little chance of seeing admission.

But if your essay topic is very original and different, you’re more likely to earn that coveted second glance at your application.

What does being unique mean exactly, though? Many students assume that they must choose an extremely rare or crazy experience to talk about in their essays —but that's not necessarily what I mean by "unique." Good college essay topics can be unusual and different, yes, but they can also be unique takes on more mundane or common activities and experiences .

For instance, say you want to write an essay about the first time you went snowboarding. Instead of just describing the details of the experience and how you felt during it, you could juxtapose your emotions with a creative and humorous perspective from the snowboard itself. Or you could compare your first attempt at snowboarding with your most recent experience in a snowboarding competition. The possibilities are endless!

#5: It Clearly Answers the Question

Finally, good college essay topics will clearly and fully answer the question(s) in the prompt.

You might fail to directly answer a prompt by misinterpreting what it’s asking you to do, or by answering only part of it (e.g., answering just one out of three questions).

Therefore, make sure you take the time to come up with an essay topic that is in direct response to every question in the prompt .

Take this Coalition Application prompt as an example:

What is the hardest part of being a teenager now? What's the best part? What advice would you give a younger sibling or friend (assuming they would listen to you)?

For this prompt, you’d need to answer all three questions (though it’s totally fine to focus more on one or two of them) to write a compelling and appropriate essay.

This is why we recommend reading and rereading the essay prompt ; you should know exactly what it’s asking you to do, well before you start brainstorming possible college application essay topics.

53 College Essay Topics to Get Your Brain Moving

In this section, we give you a list of 53 examples of college essay topics. Use these as jumping-off points to help you get started on your college essay and to ensure that you’re on track to coming up with a relevant and effective topic.

All college application essay topics below are categorized by essay prompt type. We’ve identified six general types of college essay prompts:

Why This College?

Change and personal growth, passions, interests, and goals, overcoming a challenge, diversity and community, solving a problem.

Note that these prompt types could overlap with one another, so you’re not necessarily limited to just one college essay topic in a single personal statement.

• How a particular major or program will help you achieve your academic or professional goals
• A memorable and positive interaction you had with a professor or student at the school
• Something good that happened to you while visiting the campus or while on a campus tour
• A certain class you want to take or a certain professor you’re excited to work with
• Some piece of on-campus equipment or facility that you’re looking forward to using
• Your plans to start a club at the school, possibly to raise awareness of a major issue
• A study abroad or other unique program that you can’t wait to participate in
• How and where you plan to volunteer in the community around the school
• An incredible teacher you studied under and the positive impact they had on you
• How you went from really liking something, such as a particular movie star or TV show, to not liking it at all (or vice versa)
• How yours or someone else’s (change in) socioeconomic status made you more aware of poverty
• A time someone said something to you that made you realize you were wrong
• How your opinion on a controversial topic, such as gay marriage or DACA, has shifted over time
• A documentary that made you aware of a particular social, economic, or political issue going on in the country or world
• Advice you would give to your younger self about friendship, motivation, school, etc.
• The steps you took in order to kick a bad or self-sabotaging habit
• A juxtaposition of the first and most recent time you did something, such as dance onstage
• A book you read that you credit with sparking your love of literature and/or writing
• A school assignment or project that introduced you to your chosen major
• A glimpse of your everyday routine and how your biggest hobby or interest fits into it
• The career and (positive) impact you envision yourself having as a college graduate
• A teacher or mentor who encouraged you to pursue a specific interest you had
• How moving around a lot helped you develop a love of international exchange or learning languages
• A special skill or talent you’ve had since you were young and that relates to your chosen major in some way, such as designing buildings with LEGO bricks
• Where you see yourself in 10 or 20 years
• Your biggest accomplishment so far relating to your passion (e.g., winning a gold medal for your invention at a national science competition)
• A time you lost a game or competition that was really important to you
• How you dealt with the loss or death of someone close to you
• A time you did poorly in a class that you expected to do well in
• How moving to a new school impacted your self-esteem and social life
• A chronic illness you battled or are still battling
• Your healing process after having your heart broken for the first time
• A time you caved under peer pressure and the steps you took so that it won't happen again
• How you almost gave up on learning a foreign language but stuck with it
• Why you decided to become a vegetarian or vegan, and how you navigate living with a meat-eating family
• What you did to overcome a particular anxiety or phobia you had (e.g., stage fright)
• A history of a failed experiment you did over and over, and how you finally found a way to make it work successfully
• Someone within your community whom you aspire to emulate
• A family tradition you used to be embarrassed about but are now proud of
• Your experience with learning English upon moving to the United States
• A close friend in the LGBTQ+ community who supported you when you came out
• A time you were discriminated against, how you reacted, and what you would do differently if faced with the same situation again
• How you navigate your identity as a multiracial, multiethnic, and/or multilingual person
• A project or volunteer effort you led to help or improve your community
• A particular celebrity or role model who inspired you to come out as LGBTQ+
• Your biggest challenge (and how you plan to tackle it) as a female in a male-dominated field
• How you used to discriminate against your own community, and what made you change your mind and eventually take pride in who you are and/or where you come from
• A program you implemented at your school in response to a known problem, such as a lack of recycling cans in the cafeteria
• A time you stepped in to mediate an argument or fight between two people
• An app or other tool you developed to make people’s lives easier in some way
• A time you proposed a solution that worked to an ongoing problem at school, an internship, or a part-time job
• The steps you took to identify and fix an error in coding for a website or program
• An important social or political issue that you would fix if you had the means

How to Build a College Essay in 6 Easy Steps

Once you’ve decided on a college essay topic you want to use, it’s time to buckle down and start fleshing out your essay. These six steps will help you transform a simple college essay topic into a full-fledged personal statement.

Step 1: Write Down All the Details

Once you’ve chosen a general topic to write about, get out a piece of paper and get to work on creating a list of all the key details you could include in your essay . These could be things such as the following:

• Emotions you felt at the time
• Names, places, and/or numbers
• Dialogue, or what you or someone else said
• A specific anecdote, example, or experience
• Descriptions of how things looked, felt, or seemed

If you can only come up with a few details, then it’s probably best to revisit the list of college essay topics above and choose a different one that you can write more extensively on.

Good college essay topics are typically those that:

• You remember well (so nothing that happened when you were really young)
• You're excited to write about
• You're not embarrassed or uncomfortable to share with others
• You believe will make you positively stand out from other applicants

Step 2: Figure Out Your Focus and Approach

Once you have all your major details laid out, start to figure out how you could arrange them in a way that makes sense and will be most effective.

It’s important here to really narrow your focus: you don’t need to (and shouldn’t!) discuss every single aspect of your trip to visit family in Indonesia when you were 16. Rather, zero in on a particular anecdote or experience and explain why and how it impacted you.

Alternatively, you could write about multiple experiences while weaving them together with a clear, meaningful theme or concept , such as how your math teacher helped you overcome your struggle with geometry over the course of an entire school year. In this case, you could mention a few specific times she tutored you and most strongly supported you in your studies.

There’s no one right way to approach your college essay, so play around to see what approaches might work well for the topic you’ve chosen.

If you’re really unsure about how to approach your essay, think about what part of your topic was or is most meaningful and memorable to you, and go from there.

Step 3: Structure Your Narrative

• Beginning: Don’t just spout off a ton of background information here—you want to hook your reader, so try to start in the middle of the action , such as with a meaningful conversation you had or a strong emotion you felt. It could also be a single anecdote if you plan to center your essay around a specific theme or idea.
• Middle: Here’s where you start to flesh out what you’ve established in the opening. Provide more details about the experience (if a single anecdote) or delve into the various times your theme or idea became most important to you. Use imagery and sensory details to put the reader in your shoes.
• End: It’s time to bring it all together. Finish describing the anecdote or theme your essay centers around and explain how it relates to you now , what you’ve learned or gained from it, and how it has influenced your goals.

Step 4: Write a Rough Draft

By now you should have all your major details and an outline for your essay written down; these two things will make it easy for you to convert your notes into a rough draft.

At this stage of the writing process, don’t worry too much about vocabulary or grammar and just focus on getting out all your ideas so that they form the general shape of an essay . It’s OK if you’re a little over the essay's word limit — as you edit, you’ll most likely make some cuts to irrelevant and ineffective parts anyway.

If at any point you get stuck and have no idea what to write, revisit steps 1-3 to see whether there are any important details or ideas you might be omitting or not elaborating on enough to get your overall point across to admissions officers.

Step 5: Edit, Revise, and Proofread

• Sections that are too wordy and don’t say anything important
• Irrelevant details that don’t enhance your essay or the point you're trying to make
• Parts that seem to drag or that feel incredibly boring or redundant
• Areas that are vague and unclear and would benefit from more detail
• Phrases or sections that are awkwardly placed and should be moved around
• Areas that feel unconvincing, inauthentic, or exaggerated

Start paying closer attention to your word choice/vocabulary and grammar at this time, too. It’s perfectly normal to edit and revise your college essay several times before asking for feedback, so keep working with it until you feel it’s pretty close to its final iteration.

This step will likely take the longest amount of time — at least several weeks, if not months — so really put effort into fixing up your essay. Once you’re satisfied, do a final proofread to ensure that it’s technically correct.

Step 6: Get Feedback and Tweak as Needed

After you’ve overhauled your rough draft and made it into a near-final draft, give your essay to somebody you trust , such as a teacher or parent, and have them look it over for technical errors and offer you feedback on its content and overall structure.

Use this feedback to make any last-minute changes or edits. If necessary, repeat steps 5 and 6. You want to be extra sure that your essay is perfect before you submit it to colleges!

Recap: From College Essay Topics to Great College Essays

Many different kinds of college application essay topics can get you into a great college. But this doesn’t make it any easier to choose the best topic for you .

In general, the best college essay topics have the following qualities :

• They’re specific
• They show who you are
• They’re meaningful to you
• They’re unique
• They clearly answer the question

If you ever need help coming up with an idea of what to write for your essay, just refer to the list of 53 examples of college essay topics above to get your brain juices flowing.

Once you’ve got an essay topic picked out, follow these six steps for turning your topic into an unforgettable personal statement :

• Write down all the details
• Figure out your focus and approach
• Structure your narrative
• Write a rough draft
• Edit, revise, and proofread
• Get feedback and tweak as needed

And with that, I wish you the best of luck on your college essays!

What’s Next?

Writing a college essay is no simple task. Get expert college essay tips with our guides on how to come up with great college essay ideas and how to write a college essay, step by step .

You can also check out this huge list of college essay prompts  to get a feel for what types of questions you'll be expected to answer on your applications.

Want to see examples of college essays that absolutely rocked? You're in luck because we've got a collection of 100+ real college essay examples right here on our blog!

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Hannah received her MA in Japanese Studies from the University of Michigan and holds a bachelor's degree from the University of Southern California. From 2013 to 2015, she taught English in Japan via the JET Program. She is passionate about education, writing, and travel.

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The Best Writing on Mathematics 12

Mircea pitici,  series editor.

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

The year’s finest mathematical writing from around the world

The year's finest mathematical writing from around the world

The year's finest mathematics writing from around the world

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

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

The year’s most memorable writing on mathematics

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6 Stellar Stanford Essay Examples

What’s covered:, essay example #1 – letter to your future roommate, one-second videos, essay example #2 – letter to your future roommate, study and fun, essay example #3 – letter to your future roommate, k-pop and food, essay example #4 – something meaningful, 1984, essay example #5 – something meaningful, ramen, essay example #6 – significant challenge short answer, where to get your stanford essays edited.

Stanford is one of the most selective colleges in the nation, with an acceptance rate typically under 5%. If you want to snag a spot at this renowned university in sunny California, you’ll need to write standout essays.

Stanford is known for it’s short and whimsical prompts that give students a lot of freedom to let their creativity shine through. In this post, we will be going over three essays real students have submitted to Stanford to give you an idea of how to approach your essays. We will also share what each essay did well and where there is room for improvement.

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 Stanford essay breakdown to get a comprehensive overview of this year’s supplemental prompts. 

Prompt: Virtually all of Stanford’s undergraduates live on campus. Write a note to your future roommate that reveals something about you or that will help your roommate—and us—get to know you better. (100-250 words)

Hey roomie!

I’m so excited to meet you and share our first year at Stanford, but I should probably warn you. By the end of fall quarter, I guarantee that you will be sick of hearing me ask, “Do you want to be in my one second?”

For the past couple of years, recording a one-second video every day has been my way of finding excitement in even the most boring days. I promise that while we’re roommates, my one-second clips will make every day an adventure.

Some of my personal favorites:

• Ice skating in Millennium Park in Chicago
• Watching Netflix with my 3 sisters (usually Jane the Virgin)
• Baking a cake in physics class
• Petting my 17-pound rabbit, or my 2-pound rabbit
• Family karaoke night featuring the High School Musical soundtrack and my terrible singing 
• Playing in Pep Band at basketball games with my best friends
• Winning Mario Kart (I am a self-proclaimed professional)
• Playing with a friend’s new puppy
• Selfies with my Target coworkers after handling an army of coupon moms

I’m excited to capture our first year together at Stanford, from Big Game to our first ski trip. Even on days where studying in our dorm seems like the highlight, I’ll suggest a spontaneous ice cream run so we’re not THAT lame.

So when I inevitably ask you to be in my one second, I promise that it’ll be worth it (and you can’t say I didn’t warn you).

Sincerely, 

Your soon-to-be bestie/adventure buddy/one-second-a-day-video-taking roommate

What The Essay Did Well

This is such a fun essay to read because it shows us who this student is outside of her academics and extracurriculars. There isn’t a single mention of her academic interests or the clubs and organizations she is in—ironically, that’s the strength of the essay! By focusing her essay around her one second a day video, it allows her to demonstrate to the reader her most natural self. Outside the confines of a classroom or pursuing extracurricular achievement, these are the things that bring her joy and make her interesting; conveying that idea is the exact point of Stanford asking this question.

Bulleting her most memorable one second videos is a great way to share a wide variety of stories without making the essay too dense. They are quick thoughts—not even fully formed sentences—but they all start with a verb to bring a sense of action to the essay. Not to mention, she was able to work in a good amount of humor. Including her “terrible singing ” at karaoke night, being a “ self-proclaimed professional ” at Mario Kart, and the “ army of coupon moms ” at her job isn’t necessary for each story, but adding it in gives admissions officers an extra little chuckle.

No space is wasted in this essay, even down to the sign-off. She could have ended by saying “ Sincerely, Sara “, but instead, she added an extra line to excitedly describe herself as “ Your soon-to-be bestie/adventure buddy/one-second-a-day-video-taking roommate.”  As if we didn’t get enough of a taste of her personality throughout, this student closes with a run-on thought that conveys her child-like enthusiasm at going to Stanford and meeting her roommate. 

What Could Be Improved

Overall, this is a really strong essay. That being said, there are a few sentences that could be reworked to be a bit more fun and align better with the rest of the essay.

For example, the starting off with an admission that her roommate might get sick of hearing about her one second videos is cute, but it could be made stronger by really leaning into it. “ Hi roomie! Here’s to hoping you aren’t ready to throw my phone out the third-floor window of Branner by finals!”  With this opening, we are immediately asking ourselves what could this student possibly be doing with her phone that would cause her roommate to chuck it out a window. It builds suspense and also adds humor. Not to mention, she would be including a dorm on campus to show she has thoroughly research life at Stanford.

Another sentence that could use some extra TLC is “ I promise that while we’re roommates, my one-second clips will make every day an adventure.”  Again, a nice sentiment, but it doesn’t stimulate the reader’s mind in the same way an example would. She goes into some of the one seconds they will capture at Stanford later on, but it wouldn’t hurt to add another example here. She could write something like this: “ With me everyday will be an adventure; I’ll have the clip of you trying scrambled eggs and strawberries at the dining hall for proof (trust me, it’s how they were meant to be eaten). “

Dear stranger (but hopefully future roomie),

Are you looking for someone that:

S ees you only at night when they are going to sleep?

T hrives being taciturn?

U nnerves you on the eve of your exams?

D oesn’t tell Moroccan fairy tales each night?

Y owls while sleeping?

A bhors lending you their clothes?

N ever nibbles on snacks and won’t bring you Moroccan cookies?

D oesn’t ask you to go for a walk on campus?

F idgets when you need help?

U proots a spider they cross without asking you for help?

N ot ready to sing with you if you play Beyonce’s songs?

Don’t fret if you said no to all of the above. That just means we are the perfect match because I am the opposite of everything I described above! It would be my great pleasure to introduce you to the person with whom you will not just share a room, but also have unforgettable moments. Be ready to spend nights laughing–it is not my fault if I keep you up all night with my jokes. Words cannot express how excited I am to find out what makes you, you! I’ve cleverly hidden our theme within my note. In case you didn’t notice, reread the first letter of each line.

P.S: It may be difficult for you to say the “kh” in my name, especially if you don’t speak Arabic or Spanish. So feel free to call me Yara.

This is a charming way to introduce yourself to a future roommate. Not only did they spell out all the ways they will be a loyal and dependable roommate, but they literally spelled out a secret message! Accomplishing this shows this student took extra time and care into crafting statements to add an extra layer of creativity.

This student also imbued aspects of their personality in these statements—once you flip it around. We see how important their Moroccan heritage is, as they look forward to sharing “ Moroccan fairytales each night ” and “ Moroccan cookies ” with their roommate. We see how caring they are when it comes to  “lending you clothes”  and not fidgeting “ when you need help. ” They also include some humor in some lines: “Yowls while sleeping.” Each sentence helps piece together different aspects of this student’s personality to help us put together a full picture.

Although the idea of presenting a bunch of contradictory statements puts a nice spin on the structure, be cautious about going this route if it gets too confusing for your reader. Certain lines create double negatives—” doesn’t tell Moroccan fairytales ,” “ never nibbles on snacks ,” “ not ready to sing with you “—that take the reader an extra second to wrap their head around what the student is actually trying to say. Admissions officers spend a very limited amount of time on each essay, so you don’t want to include any language that requires additional brain power to digest.

This essay is also missing the closing to the letter. The author includes “ Dear stranger ” and “ P.S. “, indicating they are writing the essay in the format of a letter. Their letter requires a closing statement and a sign-off of their name. Without them signing their name at the end of the essay, the P.S. they include doesn’t make as much sense. If the reader doesn’t know what their name is, how would they understand their nickname? 

Hey, future roommate!

As an INFJ personality type, I value my relationships and genuinely want to know you better:

How do you feel about music? I. Love. Music. My favorite genre is kpop, and since I am an avid kpop lover, I follow many groups (TXT and Twice being my favorites). I apologize in advance if you hear me blasting songs. Admittedly, getting lost in my own little world happens a lot. You can just ask me to tone it down. Or join in!

I am also a sucker for dramas. We could watch sweet heart aching love stories or historical ones together! Both are also my cup of tea.

Speaking of tea, what is your favorite drink to order? I tend to prefer sweet, bitter coffee and teas. I also like trying out new foods and making them. You know…you could be my taste tester. I like to consider myself an amateur cook. If we somehow miss the dining hours, no need to worry. With my portable bunsen stove, we can make hot pot in the dorm or quickly whip something up suitable to both our tastes.

As much as I love all food, Burmese food holds a special place in my heart. I would like to share with you my favorite foods: lahpet thoke (tea leaf salad) and ohn no khao swè (coconut noodle soup). Food is my love language, and I hope that we can share that same connection through exchanging and trying out new foods!

This essay packs a ton of information into just a few paragraphs. We learn about the author’s food and drink preferences, music taste, and favorite TV shows. The vivid language about food, drink, and cooking in particular makes the images of this student’s potential life at Stanford that much clearer and more compelling. 

Another especially strong element of this essay is the author’s personality and voice, which come through loud and clear in this essay. Through varied sentence structure and the way they phrase their stories, we get a great sense of this applicant’s friendliness and happy, enthusiastic style of engaging with their peers. 

Finally, college applications are by their nature typically quite dry affairs, and this kind of prompt is one of the few chances you might have to share certain parts of your personality that are truly essential to understanding who you are, but don’t come across in a transcript or activities list. This student does a great job taking advantage of this opportunity to showcase a truly new side of them that wouldn’t come across anywhere else in their application.

You wouldn’t, for example, want to just rehash all the APs you took or talk about being captain of your sports team. Firstly, because those probably aren’t the first things you’d talk about with your new roommate, and secondly, because that information doesn’t tell admissions officers anything they don’t already know. Instead, approach this prompt like this student did, and discuss aspects of who you are that help them understand who you are on a day to day basis—as the prompt itself hints at, the residential college experience is about much more than just class.

This is a great letter to a future roommate, but it’s important to remember that while the prompt is officially for future roommates, the essay is actually going to admissions committees. So, you want to  think carefully about what kinds of practices you mention in your essays. In most college dorms, students aren’t even supposed to light candles because it’s a fire hazard. So, while your dorm cooking skills might be very impressive, it’s probably not a good idea to advertise a plan to bring a portable stove to campus, as these kinds of things are often against dorm rules.

This may seem like nitpicking, but at a school as competitive as Stanford, you want to be extra careful to avoid saying anything that admissions officers might find off-putting, even subconsciously. For a more extreme example, you obviously wouldn’t want to talk about all the parties you plan on hosting. While this slip-up is much more minor, and the student was clearly well-intentioned, the overall genre of disregard for the rules is the same, and obviously not something you want to highlight in any college application.

Prompt: Tell us about something that is meaningful to you and why. (100-250 words)

I am an avid anti-annotationist; the mere idea of tainting the crisp white pages of any novel with dark imprints of my own thoughts is simply repulsive. However, I have one exception — my copy of George Orwell’s 1984, weathered and annotated in two languages. While victimized by uneven handwriting eating away at the margins, it is the only novel I still hold beloved despite its flaws. 

Two years before reading 1984, I was indulging in the novels of Dr. Seuss, not because of my preferences, but because my reading level was deemed an “A” — the reading level of a toddler. I was certainly anything but that; I was a fresh-off-the-plane immigrant and rising middle schooler who could barely name colors in English. 

After reading the likes of A Very Hungry Caterpillar like a madman, my next step was purchasing more advanced books in both English and Korean, so I could understand the nuance and missing details of novels after I initially read them in English. This crutch worked perfectly until George Orwell’s 1984 — the first novel I purchased and read without the training wheels of a translated copy. It took me weeks to finish the book; it was painfully slow, like a snail inching toward an arbitrary finish line. 

I read the novel twenty-seven times, each reading becoming faster and revealing more information. When I look at my copy of 1984, I still cringe at its weathered and tainted pages, but I can’t help admiring that initial portal between two literary worlds. 

This is undoubtedly an excellent writer who produced an exceptionally strong essay. Right from describing themself as an “ avid anti-annotationist, ” we can tell this is going to be different than you typical essay. While many students will choose something related to their academic or extracurricular passion, this essay choose a specific book. Although 1984 is so much more to them than simply a novel, as they reveal through the essay, the focus on an individual object as something meaningful is such a powerful image.

This student does a beautiful job conveying their journey through the symbol of 1984. They measure time using the book (“ Two years before reading 1984 “), and use well-known children’s novels like A Very Hungry Caterpillar and Dr. Seuss to convey just how far they came without explicitly needing to describe how behind they were. Describing reading 1984 without a translated copy as ditching “training wheels” further emphasizes their growth.

The meaningfulness of 1984 is reinforced through the focus on its “ weathered and tainted pages .” Admitting to the reader at the beginning that they hate marking up books, yet their favorite book is annotated from cover to cover, highlights how 1984 is so much more than a book to them. It is a symbol of their resilience, of their growth, and of a pivotal turning point in their lives. Although the student doesn’t say any of this in their essay, their skilled writing reveals all of it to the reader.

One of Stanford’s deepest values is intellectual vitality (in fact, there’s a whole separate prompt dedicated to the topic!). This student demonstrates this value through establishing a willingness to learn and a love of cross-cultural literature.  All the while, this student is authentic. There’s little posturing here intended to impress the admissions officers with the student’s resilience and deep love for the written word; instead, he is genuine in sharing a small but authentic part of his life.

This essay has very little that needs to be improved on, but there is one crucial question that would have been nice to have answered: why 1984? Out of all the books in the world, why was this the one this student decided to commit to as the first all-English novel? Was it just by chance, did a teacher encourage them to pick it up, or did the premise of the book speak to them? Whatever the reason, it would have been nice to know to further understand its significance.

While most people argue that the best invention is something mechanical or conceptual, I believe it’s the creation of instant ramen. There’s little time involvement, deliciousness, and convenience all included in one package. What more could one ask for? The nostalgia packed within instant ramen makes it a guilty pleasure I can’t live without. 

During a road trip to Yellowstone, this miracle meal followed my family as we took turns sharing an umbrella under the pouring rain and indulging it in its instant delicacy: we were shivering in the cold, but the heat of the spicy soup and the huge portion of springy noodles warmed our souls instantly. It was an unforgettable experience, and eating ramen has since then followed us to Disneyland, Crater Lake, and Space Needle, being incorporated in our frequent road trips. 

It has also come in handy during our wushu competition trips. Often, competitions ended at midnight, making it inconvenient to eat out. In these situations, the only essentials we needed were hot water and instant ramen packages, enough to satiate our spirits and hunger.

Instant ramen is also a way my mom and grandma express their care for me. On late nights of doing homework after wushu practice, I usually ate something—sometimes instant ramen—to have a smoother recovery. My mom and grandma usually paired instant ramen with extra toppings like homemade wontons or fish balls—their motto being “instant ramen always tastes better when someone makes it for you.

By picking such an unusual topic, this applicant grabs the attention and interest of readers straightaway. Picking something as commonplace and commercial as instant ramen and transforming it into a thoughtful story about family is a testament to this student’s ability to think outside the box and surprise admissions officers. It makes for an essay that’s both meaningful and memorable! 

Another great aspect of this response is how information-dense it is. We learn not just about the writer’s fondness for instant ramen, but about their family road trips, their participation in wushu, their close-knit extended family, and their culture. Even though some of these details come in the form of brief, almost throwaway lines, like briefly mentioning fishballs and wontons, they are clearly thoughtfully placed and designed to add depth and texture to the essay. 

While walking the line between maximizing every word available to you and having your essay be cohesive and easy to follow is tricky, this writer does a fantastic job of it. The details they include are all clearly relevant to their main theme of instant ramen, but also distinct enough that we get a comprehensive sense of who they are in just 250 words. Remember, even quick details can go a long way in enriching your overall description of your topic or theme.

This is a very strong essay, but there’s always room for improvement. The first paragraph of this essay, though a good general introduction that you might find in an academic essay, doesn’t actually say much about this applicant’s potential as a Stanford student. Remember, since your space is so limited in the college essay, you want every sentence, and really every word, to be teaching admissions officers something new about you.

Starting a story in media res, or in the middle of the action, can get the reader immersed in your story more quickly, and save you some words that you can then use to add details later on. Avoiding a broad overview in your first paragraph also allows you to get into the meat of your writing more quickly, which admissions officers will appreciate—remember, they’re reading dozens if not hundreds of applications a day, so the more efficient you can be in getting to your point, the better.

Everybody talks. The Neon Trees were right, everybody does indeed talk but in our society no one listens. Understandably, the inclination to be heard and understood jades our respect for others, resulting in us speaking over people to overpower them with our greatest tools, being our voices.

What The Response Did Well

This prompt is a textbook example of the “Global Issues” essay , but with an obvious catch: you have only 50 words to get your point across. With such limited space, this Stanford short answer supplement demands that applicants get their point across quickly and efficiently. This essay does a great job of grabbing one’s attention with an unusual hook that segues smoothly into the main topic. Along with that, the student demonstrates that they have a great vocabulary and sophisticated writing style in just a few sentences. 

While failing to communicate effectively indeed causes a great many problems, failure to listen is an incredibly broad challenge, and therefore, not the strongest choice for this short response. Remember, like with any other supplement, you want your response to teach Stanford admissions officers something about you. So, you ideally want to choose a specific subject that reflects both your knowledge of the world and your personal passions.

Again, your space is limited, but if this student had been even slightly more specific, we would have learned much more about their personality. For example, the sentence that starts with “understandably” could have instead read:

““Understandably, the inclination to be heard and understood jades our respect for others, which causes shortsightedness that, if nothing changes, will soon enough leave our air unbreathable and our water undrinkable.”

This version goes a step further, by not just speaking vaguely about nobody listening, but also pointing out a tangible consequence of this problem, which in turn demonstrates the student’s passion for environmentalism.

Do you want feedback on your Stanford essays? After rereading your essays countless times, it can be difficult to evaluate your writing objectively. That’s why we created our free Peer Essay Review tool , where you can get a free review of your essay from another student. You can also improve your own writing skills by reviewing other students’ essays. 

If you want a college admissions expert to review your essay, advisors on CollegeVine have helped students refine their writing and submit successful applications to top schools. Find the right advisor for you to improve your chances of getting into your dream school!

Related CollegeVine Blog Posts

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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This essay got a high-school senior into Harvard, Yale, MIT, and other top schools

Harvard University freshman Brenden Rodriquez is immersed in the strenuous course load required of his mechanical engineering major.

But before he was accepted to the prestigious Ivy League school, he had to first navigate the tricky aspect of writing a stellar admissions essay.

His hard work certainly paid off. In addition to Harvard, he gained acceptances at Yale, MIT, Columbia and the University of Virginia.

Rodriquez brilliantly merged two of his passions — music and math — to explain how each has shaped his life and improved his happiness.

Related stories

Rodriguez' other impressive stats are included on his Admitsee profile . AdmitSee is an education startup that has 60,000 profiles of students who have been accepted into college. In addition to admissions essays, and test scores, the students list other data points for prospective students to browse.

Rodriquez graciously shared his admissions essay with Business Insider, which we've reprinted verbatim below.

I think about the converging waves of the notes I play, the standing waves being created by plucking a string, and the physics behind the air pockets being forged that eventually find a listening ear whenever I sit down to play my bass. Thus, my passions of math and music synergistically become more together than they could ever be apart. I started thinking about this when a former math teacher of mine approached me one afternoon and asked me if I was interested in giving the induction speech at the Mu Alpha Theta induction ceremony. Being a member of the honor society and recounting the memorable induction speech given the year prior at my own induction, I wholeheartedly agreed. I decided on the topic of music and math because I play upright bass in the orchestra and electric bass in the jazz ensemble and being a math enthusiast, it is impossible for me not to see the mathematics and physics present in music.

At music’s core, math is present in the tempo and rhythm of a piece, with the time signature being represented as a fraction and the tempo being represented by a numerical value in beats per minute. The relationship between the two gets even more intriguing when applied to actual notes being played. The best sounding music is that which uses flawless mathematics. It is common knowledge that each note has a letter name—A through G—but also has a number value, measured in hertz. An A4 for instance is 440 hertz. In Beethoven’s “Moonlight Sonata,” there exist triads in triplet form. These triads are made up of D, F#, and A. Since sound is a vibrational energy, notes can be graphed as sine functions. When the triad notes are graphed, they intersect at their starting point and at the point 0.042. At this point the D has gone through two full cycles, the F# two and a half, and the A three. This results in consonance, something that sounds naturally pleasant to the ear. Thinking about this opened my eyes to all the aspects of my life with which I utilize math to enhance.

There is also an incredible amount of unseen math present in football. At 5 foot 10 inches and 160 pounds with pads on, I fall short of the average player at my position who is usually at least 6 feet tall and well over 200 pounds, so applying math to football is intellectually stimulating, but is more importantly a survival mechanism. When I have to go up against an opponent who is over twice my size and looks like he eats freshmen for lunch, brute force is not on my side and it helps having equations for momentum and attack angle running through my head. Math not only helps me survive, but also thrive. As an opponent running back is darting down the sideline with seemingly cheetah-like speed, I can trust that my angles and velocity will allow me to make the play and possibly save a game-changing touchdown. Or when a ball is sailing through the air caught in the stadium lights, I can picture a projectile motion problem with constant acceleration downward and a near constant velocity in the x-direction, and know that I have a leg up on the player next to me who does not think about it the way I do. When I look at aspects of my life in a math context, they make more sense and make things that I love even better and more enjoyable.

Watch: Asian-American groups are saying that affirmative action hurts their chances to get into Ivy League schools

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What Students Are Saying About the Value of Math

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

By The Learning Network

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

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

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

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

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

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

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College Algebra

Unit 1: linear equations and inequalities, unit 2: graphs and forms of linear equations, unit 3: functions, unit 4: quadratics: multiplying and factoring, unit 5: quadratic functions and equations, unit 6: complex numbers, unit 7: exponents and radicals, unit 8: rational expressions and equations, unit 9: relating algebra and geometry, unit 10: polynomial arithmetic, unit 11: advanced function types, unit 12: transformations of functions, unit 13: rational exponents and radicals, unit 14: logarithms.

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The Complete College Essay Handbook: A Step-by-Step Guide to Writing the Personal Statement and the Supplemental Essays Paperback – July 19, 2021

Want to write memorable college application essays in less time, with less stress? This book will guide you through the process, with hands-on activities, practical tips, and tons of real application essays—personal statements and supplemental essays—by real students!

Finally—the book you’ve been waiting for! The Complete College Essay Handbook demystifies the entire college essay writing process with easy-to-follow directions and hands-on activities that have worked for hundreds of students. Maschal, a former admissions officer, and Wood, a professional writer and writing teacher, draw on their combined expertise to help students craft a successful set of application essays for every school on their list. Supplemental essays in particular can seem overwhelming—some schools ask students to write as many as six essays in addition to the personal statement. Maschal and Wood identify four types of supplemental essays, walking students through how to write each one and then how to recycle these essays for other schools.

The Complete College Essay Handbook walks students through:

• What makes an essay stand out, drawing on sample essays by real students to illustrate main points
• Brainstorming activities to find the best topics for the personal statement and supplemental essays
• How to write the two central components of every application essay: scene and reflection
• Editing and revision—including techniques to cut down or expand an essay to hit the word limit
• The four types of supplemental essays and how to decode the different essay prompts, using actual essay questions
• The strategy behind a well-rounded set of application essays

The Complete College Essay Handbook is a no-frills, practical guide that will give students the confidence and know-how they need to craft the best essays for every single school on their list—in less time and with less stress.

This book is for students, high school teachers and counselors, parents, and anyone else who wants to help students through the college essay writing process.

• Print length 212 pages
• Language English
• Publication date July 19, 2021
• Dimensions 6 x 0.48 x 9 inches
• ISBN-10 173731598X
• ISBN-13 978-1737315988
• See all details

Product details

• Publisher ‏ : ‎ 1 (July 19, 2021)
• Language ‏ : ‎ English
• Paperback ‏ : ‎ 212 pages
• ISBN-10 ‏ : ‎ 173731598X
• ISBN-13 ‏ : ‎ 978-1737315988
• Item Weight ‏ : ‎ 12.2 ounces
• Dimensions ‏ : ‎ 6 x 0.48 x 9 inches
• #225 in College Guides (Books)
• #366 in College Entrance Test Guides (Books)

About the author

Brittany maschal.

Dr. Brittany Maschal is the founder of Brittany Maschal Consulting, LLC, an educational consulting firm that works with students applying to college and graduate school.

Brittany has held positions in admissions and student services at the University of Pennsylvania at Penn Law and The Wharton School; Princeton University (undergraduate) and the Woodrow Wilson School of Public and International Affairs; and the Johns Hopkins University-Paul H. Nitze School of Advanced International Studies (SAIS). She has served on admissions committees with American Councils for International Education and International Research and Exchanges Board; as an invited speaker to numerous community programs in the US and abroad; and as an alumni interviewer and admissions representative for the Graduate School of Education at the University of Pennsylvania. Brittany was also an Executive Board member and Membership Director of the Penn GSE Alumni Association.

Brittany received her doctorate in higher education from the George Washington University in 2012. Prior, she attended the University of Pennsylvania for her master’s, and the University of Vermont for her bachelor’s degree—a degree she obtained in three years. Brittany is a member of the Independent Educational Consultants Association and a member of the Professional Association of Resume Writers and Career Coaches.

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Customers find the themes in the book comprehensive and personal. They also appreciate the writing style as clear, empathic, and gentle.

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Customers find the themes in the book comprehensive, inspiring, and personal. They also say it's a gem of a book, chock full of real essays by real students. Customers also appreciate the brainstorming exercises and writing prompts.

"...was the emphasis on how students can find a meaningful and very personal story in seemingly small every day events, coupled with the very practical..." Read more

"...exercises, and writing prompts - all designed to be the perfect guided experience . It’s a gem of a book, chock full of real essays by real students!" Read more

"...This book really taught me how to write a concise , personal, meaningful college essay. So glad I found this book!" Read more

"...Not only does it help you write the common app essay , but it teaches you how to write a wide variety of supplemental essays which are just as..." Read more

Customers find the writing style clear, empathic, and gentle. They also say the book is chock full of real essays by real students.

"... It's written so clearly , and with great empathy and the right amount of gentle humor...." Read more

"...It’s a gem of a book, chock full of real essays by real students!" Read more

"... Love the clarity and concision too. I strongly recommend for students and teachers." Read more

"...The authors are very down to earth and give simple but important tips for writing a stand out essay...." Read more

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The Personal Statement Topics Ivy League Hopefuls Should Avoid

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A compelling personal statement is a critical component of an Ivy League application, as it offers students the unique opportunity to showcase their personality, experiences, and aspirations. Kickstarting the writing process in the summer can give students a critical advantage in the admissions process, allowing them more time to brainstorm, edit, and polish standout essays. However, as students begin drafting their essays this summer, they should bear in mind that selecting the right topic is crucial to writing a successful essay. Particularly for students with Ivy League aspirations, submitting an essay that is cliche, unoriginal, or inauthentic can make the difference between standing out to admissions officers or blending into the sea of other applicants.

As ambitious students embark on the college application process, here are the personal statement topics they should avoid:

1. The Trauma Dump

Many students overcome significant hurdles by the time they begin the college application process, and some assume that the grisliest and most traumatic stories will attract attention and sympathy from admissions committees. While vulnerability can be powerful, sharing overly personal or sensitive information can make readers uncomfortable and shift focus away from a student’s unique strengths. Students should embrace authenticity and be honest about the struggles they have faced on their path to college, while still recognizing that the personal statement is a professional piece of writing, not a diary entry. Students should first consider why they want to share a particular tragic or traumatic experience and how that story might lend insight into the kind of student and community member they will be on campus. As a general rule, if the story will truly enrich the admissions committee’s understanding of their candidacy, students should thoughtfully include it; if it is a means of proving that they are more deserving or seeking to engender pity, students should consider selecting a different topic. Students should adopt a similar, critical approach as they write about difficult or sensitive topics in their supplemental essays, excluding unnecessary detail and focusing on how the experience shaped who they are today.

2. The Travelogue

Travel experiences can be enriching, but essays that merely recount a trip to a foreign country without deeper reflection often fall flat. Additionally, travel stories can often unintentionally convey white saviorism , particularly if students are recounting experiences from their charity work or mission trips in a foreign place. If a student does wish to write about an experience from their travels, they should prioritize depth not breadth—the personal statement is not the place to detail an entire itinerary or document every aspect of a trip. Instead, students should focus on one specific and meaningful experience from their travels with vivid detail and creative storytelling, expounding on how the event changed their worldview, instilled new values, or inspired their future goals.

3. The Superhero Narrative

Ivy League and other top colleges are looking for students who are introspective and teachable—no applicant is perfect (admissions officers know this!). Therefore, it’s crucial that students be aware of their strengths and weaknesses, and open about the areas in which they hope to grow. They should avoid grandiose narratives in which they cast themselves as flawless heroes. While students should seek to put their best foot forward, depicting themselves as protagonists who single-handedly resolve complex issues can make them appear exaggerated and lacking in humility. For instance, rather than telling the story about being the sole onlooker to stand up for a peer being bullied at the lunch table, perhaps a student could share about an experience that emboldened them to advocate for themselves and others. Doing so will add dimension and dynamism to their essay, rather than convey a static story of heroism.

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Similarly, many students feel compelled to declare their intention to solve global issues like world hunger or climate change. While noble, these proclamations can come across as unrealistic and insincere, and they can distract from the tangible achievements and experiences that a student brings to the table. Instead, applicants should focus on demonstrable steps they’ve taken or plan to take within their local community to enact positive change, demonstrating their commitment and practical approach to making a difference. For instance, instead of stating a desire to eradicate poverty, students could describe their extended involvement in a local charity and how it has helped them to discover their values and actualize their passions.

5. The Sports Story

While sports can teach valuable lessons, essays that focus solely on athletic achievements or the importance of a particular game can be overdone and lack depth. Admissions officers have read countless essays about students scoring the winning goal, dealing with the hardship of an injury, or learning teamwork from sports. Students should keep in mind that the personal essay should relay a story that only they can tell—perhaps a student has a particularly unique story about bringing competitive pickleball to their high school and uniting unlikely friend groups or starting a community initiative to repair and donate golf gear for students who couldn’t otherwise afford to play. However, if their sports-related essay could have been written by any high school point guard or soccer team captain, it’s time to brainstorm new ideas.

6. The Pick-Me Monologue

Students may feel the need to list their accomplishments and standout qualities in an effort to appear impressive to Ivy League admissions officers. This removes any depth, introspection, and creativity from a student’s essay and flattens their experiences to line items on a resume. Admissions officers already have students’ Activities Lists and resumes; the personal statement should add texture and dimension to their applications, revealing aspects of their character, values and voice not otherwise obvious through the quantitative aspects of their applications. Instead of listing all of their extracurricular involvements, students should identify a particularly meaningful encounter or event they experienced through one of the activities that matters most to them, and reflect on the ways in which their participation impacted their development as a student and person.

7. The Pandemic Sob Story

The Covid-19 pandemic was a traumatic and formative experience for many students, and it is therefore understandable that applicants draw inspiration from these transformative years as they choose their essay topics. However, while the pandemic affected individuals differently, an essay about the difficulties faced during this time will likely come across as unoriginal and generic. Admissions officers have likely read hundreds of essays about remote learning challenges, social isolation, and the general disruptions caused by Covid-19. These narratives can start to blend together, making it difficult for any single essay to stand out. Instead of centering the essay on the pandemic's challenges, students should consider how they adapted, grew, or made a positive impact during this time. For example, rather than writing about the difficulties of remote learning, a student could describe how they created a virtual study group to support classmates struggling with online classes. Similarly, an applicant might write about developing a new skill such as coding or painting during lockdown and how this pursuit has influenced their academic or career goals. Focusing on resilience, innovation, and personal development can make for a more compelling narrative.

Crafting a standout personal statement requires dedicated time, careful thought, and honest reflection. The most impactful essays are those that toe the lines between vulnerability and professionalism, introspection and action, championing one’s strengths and acknowledging weaknesses. Starting early and striving to avoid overused and unoriginal topics will level up a student’s essay and increase their chances of standing out.

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About sophia.

Elementary (3-6) Math, Elementary (3-6) Reading

Bachelors in Education, General from James Madison University

Masters in Elementary Education and Teaching from James Madison University

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A passionate educator dedicated to cultivating young minds as a 6th-grade teacher.

I Love Tutoring Because

it allows for a personalized approach to teaching, tailored to the unique needs and learning styles of each student. Witnessing the moment when a concept clicks for a student is immensely rewarding and fuels a passion for education. The opportunity to make a positive difference in a student's academic journey while fostering confidence and a love for learning is why I love tutoring.

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