science and math essay

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How to Creatively Integrate Science and Math

Making a case for more collaboration between math and science teachers.

Why is the sky blue? I remember in my physical science class, our teacher showed us a possible reason why the sky is blue. He took a canister of liquid oxygen and poured it out on the table. I saw the blueness of the liquid as it flowed out and then disappeared. Then we talked about color, frequencies, and absorption, reflected and radiated light. I wondered how scientist ever figured these things out? Duh -- math! How can you really teach science without math? It is impossible. Science is the application of math.

In science, geometric principles such as symmetry, reflection, shape, and structure reach down to the atomic levels. In science, algebraic balance is required in chemical formulas, growth ratios, and genetic matrices. In science, math is used to analyze nature, discover its secrets and explain its existence and this is the big problem. Science is so complex and getting more so each day. In order to In math class one of the biggest needs is relevance. Students want to know how they are going to benefit from being able to do calculations. Why not use science to teach math? Since one of the biggest uses of mathematics in science is data gathering and analysis, that is the best place to start. When a teacher gives students a real science problem to solve -- one that requires math tools -- the teacher is giving the students a reason to use math. Math then becomes something useful, not something to be dreaded.

Being able to teach math better and being able to teach science better are powerful reasons for the math and science teacher collaborate with each other. According to a case study conducted by Jennifer Dennis and Mary John O'Hair, another reason that math and science teachers should collaborate is that science helps provide relevance to math that is all too often abstract and isolated calculation operations. Instead of using the word problems out of the book that many dread, why not walk down the hall and talk to the science teacher to find out what math they are using in science class? Ultimately, as another study reported, the students' increased conceptual understanding of math and science is the greatest benefit of math and science teacher collaboration. Conceptual understanding means the students know the bigger picture of why things work in math and science, not just how to make them work.

Unfortunately, knowing that increased teacher collaboration in math and science will benefit students and teachers is not enough. Teachers are so busy that finding time to collaborate is difficult. Add to this, the structure of the school inhibits collaboration when math and science teachers are spread out in a large campus. How do you overcome this? Well, a simple request to the principal might do the trick. Let the principal know what you want to do and show him the evidence and if he can, he will help. Another solution is that even though geographically speaking the math and science teachers may be isolated, everyone has a cellphone. Just call them. Texting, Facebook, Google+, Skype, Google Hangouts, or even email can bring science and math teachers together. There are many options -- just do it!

What are ways you work with your companion subject teacher (math or science) to help students understand math and science better?

Math Essay Ideas for Students: Exploring Mathematical Concepts

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

• The Beauty of Fibonacci Sequence

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

• The Mathematics of Music

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

• The Geometry of Architecture

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

• Fractals: Nature's Infinite Complexity

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

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

• The Game Theory Revolution

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

• Chaos Theory and the Butterfly Effect

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

• The Mathematics Behind Cryptography

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

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

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

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

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

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One school. Many options. Limitless possibilities.

North Carolina School of Science and Mathematics educates academically talented students to become state, national, and global leaders in science, technology, engineering, and mathematics. Founded as the nation’s first public, residential STEM high school, we let North Carolina students design your future through challenging programs driven by instructional excellence and the excitement of discovery.

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Residential programs.

Some 980 students live and learn in our Residential program on our Durham and Morganton campuses

Durham Residential

Each year, 680 students live and learn in the two-year Residential program as high school juniors and seniors taking rigorous math, science, and humanities courses at NCSSM-Durham, situated among historic in-town neighborhoods within walking distance of Duke University in this mid-size city. The campus was founded in 1980 to create the nation’s first public, residential high school featuring STEM education.

Programs At A Glance

Morganton Residential

NCSSM’s two-year Residential high school experience with rigorous math, science, and humanities courses is now offered at our newest campus composed of new and newly renovated historic buildings on a ridge-top setting. In this picturesque western North Carolina town surrounded by natural beauty and outdoor recreation, each year we serve 300 juniors and seniors from across NC in a life-changing living-learning community.

A growing number of North Carolina high school students are finding that staying enrolled in their current school while also taking challenging courses virtually at NCSSM is a great option — and one that catches the eyes of college admissions officers. NCSSM offers virtual learning options that could be the perfect way to design your future.

NCSSM Connect

Semester-long distance education courses offered virtually via videoconference during the school day at a student’s home school which is an NCSSM Connect partner school. These tuition-free, honors-level and AP courses in STEM and humanities expand the curriculum offered at North Carolina schools.

NCSSM Online

A supplemental, sequenced two-year honors program that provides the NCSSM experience virtually through evening webinars and asynchronous assignments to students across the state who remain in their home community. Students completing the program receive a certificate of completion, with the option to dual enroll for local school course credit.

Summer Programs

For rising 5th through 12th graders who seek academic challenge, growth, and like-minded peers

Accelerator

Tuition-based courses available to students from anywhere in the world that offer students academic growth through engaging, challenging STEM courses on topics they can relate to. Students learn with like-minded peers and are taught by scientists, university professors, engineers, and others who bring real-world experience and activities to the Accelerator learning experience.

Early Accelerator

Summer academic programs that offer younger students exciting and innovative ways to explore STEM concepts. Through interactive and hands-on activities, students will embark on a series of STEM adventures. These non-residential, day programs are held on NCSSM’s Durham and Morganton campuses.

INtroducing STEM Pathways through Inquiry and Research Experiences is a free, one-week residential summer program for rising 11th graders who live in NC designed for, but not limited to, students from underrepresented groups. The weeklong residential program at NCSSM-Morganton pairs students with mentors to work on project offering experiences in STEM research and in solving complex, real-world problems.

Step Up to STEM

An innovative interdisciplinary academic program that uses exciting real-world subjects to challenge rising 9th and 10th graders across North Carolina and build their skills in science, math, technology, and communications.

Summer Ventures

A a no-cost , state-funded program for academically talented North Carolina students who aspire to careers in science, technology, engineering, and mathematics. You will have the opportunity to engage in research with faculty and get hands-on experience at a university.

NC Teamship Showcase welcomes students who have participated in a Teamship Experience at schools or organizations hosting the Teamship program . This includes the many districts participating in the Spark Teamship Program.

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Durham Commencement

43rd commencement exercises, saturday, may 25, 2024, at 9 a.m..

Please join us as we graduate our 43rd class of Residential students on Saturday, May 25, on the school’s historic Durham campus. The 318 accomplished candidates represent all 13 Congressional Districts across the state of North Carolina.

Location: Watts Lawn

Morganton Commencement

1st commencement exercises, friday, may 24, 2024, at 9 a.m..

Please join us as we graduate our first class of Residential students on Friday, May 24, on the school’s Morganton campus. The 145 accomplished candidates represent all 13 Congressional Districts across the state of North Carolina.

Location:  Lawn outside of the Barn

How to apply

Eligible applicants may apply to NCSSM’s Residential, Online, or Summer Ventures program or all three, using one electronic application. There is no application fee or cost for tuition, meals, room/board, or textbooks for these three programs.

In order to complete the 2024-25 application process and be considered for the NCSSM Class of 2027, potential applicants must complete ALL of the requirements within the required time frames.

The application process for the Class of 2027 will open on Oct. 15, 2024, at noon and close on Jan. 5, 2025, at 11:59 p.m.

Be eligible

• Applicants must be in their second year of high school
• Applicants applying for the Class of 2027 must establish permanent legal residency in the state of North Carolina by December 1, 2024, and are required to complete the residency application using the North Carolina Residency Determination Services (RDS) by January 30, 2025.

Standardized tests (SAT or ACT)

Standardized test update for the class of 2027.

At this time a decision has not been determined regarding standardized testing for the Class of 2027. The NCSSM Admissions office is preparing as if standardized testing requirements will return.

Complete application

NCSSM has partnered with the  College Foundation of North Carolina (CFNC)  to create the NCSSM electronic application. To complete an application, you must have a  CFNC account . Go to  www.cfnc.org  read through the instructions and create a CFNC account. Once you have created an account or if you already have a CFNC account, log into CFNC.org.

For specific instructions on filling out the NCSSM Application at CFNC.org, see the  Application Help Guide .

Preview essay questions.

Applicants must have each section of the application completed via the CFNC.org electronic application by January 5, 2025, at 11:59 p.m. There are NO EXCEPTIONS to this deadline.

Request evaluations.

Evaluations are required from each of these by  February 28, 2025:

• 9th or 10th STEM (Science, Technology, Engineering, Math) Teacher
• 9th or 10th grade English Teacher
• Guidance counselor— Must be your current 10th grade guidance counselor

Submit a transcript

February 15 to february 28, 2025.

The window for submitting transcripts and report cards will open on February 15, 2025, and close on February 28, 2025. Applicants can submit or report their grades using one of the following methods:

• Log in to CFNC.org
• Under the Apply Section, click To College , and then  Transcript Manager
• Select  North Carolina School of Science and Math from the pull-down menu
• Applicants can request that their counselor send an official copy by US Postal Service: Attn: NCSSM Office of Admissions, 1219 Broad Street, Durham, North Carolina, 27705 or by uploading it and sending it electronically to  [email protected] .
• Self Reporting:  Applicants can upload an unofficial copy of their transcript in Blackbaud. If the applicant is enrolled in a year-long course, they will need to scan and submit a copy of their report card in Blackbaud to accompany their transcript. Note: Applicants that are extended an offer for enrollment will be contingent on NCSSM receiving an official transcript from their home high school.

NCSSM Discovery Days (NCSSM Math Assessment)

Applicants for the Class of 2027 who apply to NCSSM residential campuses (Durham and/or Morganton) are required to take the NCSSM Math Assessment as part of their application. Applicants will register in their Blackbaud account to take the NCSSM Math Assessment on one of the following dates:

Class of 2027 Discovery Days dates are coming soon. 

Note:  Applicants will receive information about registering for one of the assessment dates once the CFNC application process closes on January 5, 2025.

Helping NC students design your future

Upcoming events.

Spring Choral and Piano Concert (Morganton)

May 11, 2024

The Barn at NCSSM-Morganton

Spring Choral Concert (Durham)

NCSSM-Durham (ETC Auditorium)

Spring Wind Ensemble Concert (Durham)

May 12, 2024

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How to Write the Caltech Supplemental Essays 2023-2024

Caltech has four required supplemental essays, and three shorter optional essays, with word limits of 150, 100, and 50, respectively. Because Caltech is one of the most academically rigorous schools in the country, you want to be sure that your essays capture your intellectual and creative potential. In this post, we’ll break down each prompt so that you can know what you need to do to craft a response that truly shines.

Caltech Supplemental Essay Prompts

All applicants, required prompts.

Prompt 1: Because of the rigorous courses in the core curriculum , Caltech students don’t declare a major until the end of their first year. However, some students arrive knowing which academic fields and areas already most excite them, or which novel fields and areas they most want to explore.

If you had to choose an area of interest or two today, what would you choose? (There are dropdown menus from which you can choose up to two areas of interest.)

Why did you choose that area of interest (200 words).

Prompt 2: At Caltech, we investigate some of the most challenging, fundamental problems in science, technology, engineering, and mathematics. Identify and describe two STEM-related experiences from your high school years, either in or out of the classroom, and tell us how and why they activated your curiosity. What about them made you want to learn more and explore further? (100-200 words per experience)

Prompt 3: The creativity, inventiveness, and innovation of Caltech’s students, faculty, and researchers have won Nobel Prizes and put rovers on Mars , but Techers also imagine smaller scale innovations every day, from new ways to design solar cells to 3D printing dorm decor. How have you been an innovator in your own life? (200-250 words)

Prompt 4: Caltech’s mission – to cultivate learning, discovery, and innovation for the benefit of humanity – relies on its community members embracing fundamental Caltech values :

Openness and enthusiasm for having preconceptions challenged

Respect and appreciation for the idea that, while we are all members of the same community, the opportunities we’ve had to develop, showcase, and apply our talents have not been equal, passion for the ideal that science can and should meaningfully improve the lives of others, share what one or more of these values evokes for you. (200-400 words), all applicants, optional prompts.

Prompt 5: If there are aspects of your life or social or personal identity that you feel are not captured elsewhere in this application, please tell us about them. (150 words)

Prompt 6: When not surveying the stars, peering through microscopes, or running through marathons of coding, Caltech students pursue an eclectic array of interests that range from speed-cubing to participating in varsity athletics to reading romance novels. What is a favorite interest or hobby and why does it bring you joy? (100 words)

Prompt 7: Did you have a hard time narrowing it down to just one interest or hobby? We understand – Caltech students like to stay busy, too – tell us about another hobby or interest! (50 words)

Because of the rigorous courses in the core curriculum , Caltech students don’t declare a major until the end of their first year. However, some students arrive knowing which academic fields and areas already most excite them, or which novel fields and areas they most want to explore.

After you select your area(s) of interest, you are asked to provide the reasoning behind your choice(s). This is a fairly standard “Why This Major?” prompt . This straightforward prompt is intended to give the admissions committee a sense of what interests you, why it interests you, and why/how you plan on pursuing this interest in college and beyond.

Before we continue, we have to address the elephant in the room—what if you’re undecided?

The bad news is that you’re required to pick at least one area of interest on your application. The good news is that you aren’t contractually bound to the area you choose! In fact, “Every first-year student at Caltech takes the same classes during their first two terms; you won’t even declare your major until the end of your first year.”

Don’t worry if you haven’t figured out exactly what you want to do in college—many students haven’t! Look through the list of areas of interest and pick one that’s closely related to a hobby or pastime of yours so you’ll have something to write about.

If you do have an area of interest or desired major in mind, great! Pick that one for your essay.

Now that you’ve picked a subject, you may find it helpful to ponder the following questions before you begin crafting your response:

1) What are your sincere reasons for wanting to major in your chosen field?

Ideally, you will have picked a field in which you have a deep interest—one that you can talk about at length. You should have meaningful reasons for wanting to pursue your chosen field. If your primary motivation involves money, status, or pressure from your parents, you’re already off to a bad start. An essay that seems disingenuous or too self-serving will detract immensely from your application as a whole, so be sure to choose substantial reasons.

2) What are some specific examples of things you enjoy about this field of study?

When answering this question, aim to be as specific as you can. Anyone can write about liking “information and data sciences” or “biology,” so think of more narrow subtopics like “principal component analysis to reduce dataset dimensionality” or “identifying mitotic mutations in fruit flies.” If you’ve picked a topic you’re already passionate about and familiar with, talking about something specific you enjoy about it shouldn’t be too daunting.

3) How does this major serve your life and/or career goals?

You might not have the most detailed plan for your career and adult life, and that’s totally fine! However, it would be helpful if you had some idea of what you want to do in the future. Think of industries you would be able to work in with a degree in your chosen field. What is your dream job? How can this major contribute to your attainment of that job and success in the field?

4) What’s your favorite experience with this subject in school? What are the best parts of your experience with it outside the classroom?

5) Is there any recurring emotion or state of mind that you experience when exploring this field of study? What do you find appealing about that emotion or state of mind?

You can use your answers to questions 4 and 5 to recall some relevant anecdotes that may contribute to your response.

Once you’ve figured out the answers to the five aforementioned questions, you can begin planning a structure for your response. You may find it helpful to break your essay into two principal parts:

• The experiences that fostered and increased your interest in this field (as well as your emotional and personal connection to your chosen major)
• What you hope to do in the future, both at Caltech and in your career

Now, you should do some research on Caltech’s website to find some unique aspects of your chosen major that you can write about. Check out Caltech’s list of majors , as it has links that will lead you to each major’s respective webpage. Also consult their lists of faculty members and research facilities to see what work Caltech is doing in your area of interest.

For example, consider a hypothetical student who wants to pursue bioengineering with a particular interest in stem cell research. She might begin with an anecdote about how her father was a participant in a clinical trial for stem cell therapy after his spinal cord treatment. Perhaps seeing the potential of stem cell treatment opened up a new world for her, which fostered a deeper interest in biology and bioengineering than she had ever had before.

She might write about her high school experiences with biology classes, her intensive preparation for the AP Biology exam, and the bioengineering publications she now likes to read in her free time. She can then transition into a discussion of what kind of research she would like to be a part of at Caltech. A faculty member she can talk about specifically is Professor Magdalena Zernicka-Goetz, whose lab used stem cells last year to create model mouse embryos “that have beating hearts, as well as the foundations for a brain and all of the other organs in the mouse body.”

No matter how unique, weird, or quirky you think your interests are, there will probably be a major or research group at Caltech that can cultivate them. Don’t be afraid to show how unique you are as an individual—that’s the point of supplemental essays!

At Caltech, we investigate some of the most challenging, fundamental problems in science, technology, engineering, and mathematics. Identify and describe two STEM-related experiences from your high school years, either in or out of the classroom, and tell us how and why they activated your curiosity. What about them made you want to learn more and explore further? (100-200 words per experience)

This prompt is supposed to gauge your interest in and experiences with STEM, both in school and in your personal life. Writing a successful essay will ideally show the admissions committee a few things:

• You are serious about pursuing STEM in college and beyond.
• You have hands-on experience in STEM.
• You have at least some idea of what to expect from a STEM-based education at Caltech.

You’ll probably know if an experience is related to STEM at first glance. Nevertheless, before you begin writing your responses to this prompt, you should make sure you have a handle on what exactly STEM is, even if you think you understand it fully.

As a quick refresher, STEM stands for Science, Technology, Engineering, and Math. It heavily emphasizes analytical and critical thinking skills, scientific literacy, and domain-specific hard skills that are essential to many career paths.

Even though the prompt specifies that you can write about experiences both within and outside the classroom, you might find the tips in CollegeVine’s guide to the extracurricular activities essay helpful.

If you have any obvious STEM experience, picking some events should be fairly straightforward. Think about the experiences you’ve had with science fairs, robotics clubs, biology or chemistry classes, etc. Narrow down your experiences to the ones that had the most significant impact on your interest in STEM. If you write about an experience that you didn’t have too much emotional investment in, you might inadvertently express a tepid interest in STEM as a whole.

If you only took one science class in high school and didn’t participate in any STEM-related extracurriculars, don’t fret! Feel free to write about two experiences from the same class. You might even be able to relate some work experience to STEM.

For example, perhaps you worked alongside a pharmacist during high school. Some people consider pharmacy to be more medical and less STEM-related, but the field of pharmacology is indisputably a branch of biology and chemistry. Don’t be afraid to bend some definitions when identifying meaningful STEM experiences you’ve had.

To help you identify your two experiences, mull over a few questions:

1) What is your favorite STEM-related activity? If you don’t have a good STEM activity, which of your non-STEM activities can be linked to STEM logically?

2) What about this activity generated your interest in STEM? Why did it make you curious and how did your participation in it increase your interest?

3) What went through your mind as you participated in this activity?

4) Have you developed or strengthened any specific interests because of this activity? If so, what are they and how have they changed over time? For example, you might have been interested in chemistry in general, but this particular activity focused your attention on metal alloys.

5) Are there any specific STEM-related skills that you have developed as a result of participating in this activity? Think about hard skills like chemical titration, building robots, testing the pH of substances, etc.

Once you’ve decided on your two activities, you can begin writing your responses. For each activity, you should address each point of the prompt:

• How did the activity activate your curiosity?
• Why did the activity activate your curiosity?
• What about the activity made you want to learn more and explore further?

After addressing each point, if you still have room within your 200 words per activity, you can explain some things further. Perhaps you want to discuss something specific that you learned or exactly what your role in the activity was.

For example, consider the following response by a hypothetical student:

“During my junior year of high school, I joined the Robotics Club with no prior experience, other than having taken AP CompSci. Our team’s first project involved building and coding a robot that could get to distant water sources, collect water, and purify and store it. We spent several weekends and late nights programming the bot and troubleshooting it.

It had trouble navigating at first, then it failed to recognize the water sources. Finally, we completed the build in a few months, and though it was far from perfect, it did the job. That first drink of purified water from the bot was beyond refreshing.

That project was the one that truly showed me how useful robotics could be to humanity. I imagined building hundreds of robots like the original and sending them to developing countries to increase their access to clean drinking water. I am now drawn to mechanical engineering because it offers limitless opportunities to create devices that can be used to improve the world and people’s quality of life.”

This example is effective because it tells an engaging anecdote, addresses each point of the prompt, and offers a plan for the student’s college career and future professional life.

The creativity, inventiveness, and innovation of Caltech’s students, faculty, and researchers have won Nobel Prizes and put rovers on Mars , but Techers also imagine smaller scale innovations every day, from new ways to design solar cells to 3D printing dorm decor. How have you been an innovator in your own life? (200-250 words)

This prompt is trying to determine how you think as a problem solver. The admissions committee wants to know the ways in which you have been innovative or have approached problems creatively.

Don’t feel like you have to have developed some revolutionary solution to a global problem. No one is expecting you to have engineered some brilliant scientific apparatus—you haven’t even begun college yet! Just like the prompt’s examples illustrate, you can think on as big or as small a scale as you’d like.

When trying to choose an example of innovation in your own life, it might be useful to think about abstract qualities then work your way to concrete events. Are you ambitious, adaptable, creative, resourceful, open-minded? What are some positive qualities of yours? Once you’ve decided on some attributes, you should be able to relate them to some anecdotes.

Let’s say you’re creative and resourceful. Think of a time when those traits generated something innovative or novel in your life. Maybe you were locked out of your apartment and used a credit card to open the latch. Perhaps this experience inspired you to 3D print a plastic card to use specifically for problems like that.

Your story of innovation can involve anything really, as long as you came up with a creative solution to a problem you were confronted with. Maybe the arm of your glasses broke in the middle of class, so you attached a pen cap to it so the glasses could still be used until you had time to replace them.

You have 250 words, so you may want to think of 2-3 anecdotes to discuss. It might even be helpful to write about something you want to improve but haven’t yet. You can talk about a persistent problem you’ve seen and propose a creative potential solution.

Here’s an example of an anecdote a student might write:

“I worked at a hardware store during high school to help my parents pay some bills. The store was far from my home, and often took about an hour to get to by public transportation. I’ve always been interested in tinkering with mechanical devices, so I decided to make the most of my job.

I bought parts from the store with portions of my paycheck every couple of weeks, and over the course of several months I built myself a bicycle from scratch. Of course, it wasn’t the most visually appealing or comfortable bike, but it did the job and it did it well. I don’t use it much anymore, but I still own it and feel great pride every time I pass it on my way out the door.”

This is a good anecdote because it presents an issue, describes something about the student’s creative and inquisitive nature, and showcases the innovative solution that the student devised.

You may want to approach this prompt in a similar way. Outline a problem you had to deal with, describe some relevant positive attributes about yourself, then explain how these attributes helped you find an innovative solution to the problem.

Caltech’s mission – to cultivate learning, discovery, and innovation for the benefit of humanity – relies on its community members embracing fundamental Caltech values :

Brainstorming your topic:

Caltech has narrowed your list of possible topics to just three things—the values listed above. Now, that doesn’t mean your brainstorming process is over when you pick the value(s) you want to write about.

You also want to have a clear sense of how you’re going to explain what that value means to you, as 400 words is on the longer side for a supplemental essay. If you’re unfocused going into the essay, your writing may end up somewhat scattered. To ensure that doesn’t happen, think of some experiences you’ve had that showcase what the value you’ve selected means to you.

For example, you might want to write about your openness to other perspectives. Maybe you describe a snowshoeing trip you took with your family, and how you were dreading the damp, the cold, and the blisters. But, even though you did end up confronting all of those things, you also unexpectedly got to see the northern lights. That once-in-a-lifetime treat helped you realize that having a positive outlook on new experiences is a choice, and if you consciously make it, you’re likely to enjoy yourself much more.

As this example illustrates, unless you choose the third value, your response doesn’t have to rely solely on STEM-related experiences. Obviously, Caltech is one of the most prestigious tech schools in the world, but remember that admissions officers will also be reading your responses to Prompts 1, 2, and 3, which are all academically focused. So, if you want to share a slightly different side of yourself, this prompt is a great opportunity to do so.

Note: given the linked webpage and the wording of the values in the prompt, you might use this space to write about a topic related to diversity . Just be sure to follow the prompt’s directions if you do.

Finally, note that the prompt says you can write about “one or more” of the listed values. If you can think of an experience you’ve had that showcases two or three of the given values, go for it! That said, don’t force anything. A well-written, cohesive response that focuses on just one value is just as good as one that includes multiple values.

400 words should be more than enough to develop your ideas in sufficient detail, but if you’re trying to cram in a connection that isn’t really logically there, your essay will feel disjointed.

Tips for writing your essay:

Once you start writing, keep this important writing principle in mind—show, don’t tell. You don’t want to just state things in a factual, direct way. Rather, describe a situation that illustrates the points you’re trying to make. To see the difference, compare the following two example sentences:

Example 1: “Although I had been having a terrible time all day, when we finally reached the overlook we had been trekking towards, we got to see the northern lights, which I will never forget.”

Example 2: “When we finally emerged from the treeline, my hair was still damp with cold sweat and snowfall, and my heels were still chafing against my itchy wool socks, but my discomfort melted away in the iridescent shimmer of the aurora borealis sparkling on unblemished snow.”

These two examples are about the same moment, but the second immerses us in the scene with sensory details and strong descriptions, which makes it much more engaging to read. Since you have a little more room to write in this supplement, don’t be afraid to show your creative writing abilities.

Also be sure you give the admissions officers a strong sense of what your chosen value means to you. Remember, within your application as a whole, the point of the essay is to set yourself apart even from other smart, talented applicants. Since everyone is going to be writing about the same three given values in their responses, make sure it’s clear how your experiences have shaped your own personal understanding of the value you select. 

Including a personal overall takeaway will help admissions officers see why the experience you’ve described speaks to the kind of Caltech student you’ll be. Remember, their job is to visualize how you will contribute to their school for the next four years, so make sure you explain how something that took place in the past continues to influence you today.

Mistakes to avoid:

The most important thing to avoid in your response is vagueness. If you speak only in general abstract terms about the value you’ve selected and fail to incorporate detailed, illustrative examples, your essay may end up sounding preachy, or like a Hallmark card.

The other, slightly more subtle mistake you want to watch out for is not drawing an obvious connection between your experiences and the value you’ve selected. For example, say that in the snowshoeing example you spent all 400 words describing how beautiful the northern lights are, and didn’t say anything about how grumpy you were at the start of the snowshoeing trip. Your reader wouldn’t have any idea how this story reflects your open-mindedness.

As long as you follow the brainstorming tips we’ve provided above and rely on strong descriptions once you start writing, you should be unlikely to fall into these traps and should be well on your way to a personal, engaging essay.

Prompt 5 (Optional)

If there are aspects of your life or social or personal identity that you feel are not captured elsewhere in this application, please tell us about them. (150 words).

While we typically encourage students to respond to optional prompts, there’s no need to write additional information here if you feel that your application already captures your identity adequately.

This is a variation on the common diversity prompt . Unlike the previous prompts, this prompt has a 150-word limit, so if you choose to respond to it, you’re going to have to be more succinct.

Think about communities that you’re a member of, especially those that have played a role in your development as a person. These communities can be physical environments, like the neighborhood you grew up in, or communities defined by attributes, like your ethnic group or gender identity. Remember, identity encompasses a wealth of attributes.

Aspects of identity include traditional markers of diversity, such as ethnicity/race, country of origin, gender identity, sexual orientation, your first language, and an illness/disability. However, aspects of identity also include your hometown, socioeconomic class, groups you’re part of, and even your interests or hobbies.

A quick note if you intend to write about your racial background: In June 2023, the United States Supreme Court struck down the use of affirmative action in college admissions. The ruling, however, still allows colleges to consider race on an individual basis, which is one reason many schools are now including diversity prompts as one of their supplemental essay prompts. If you feel that your racial background has impacted you significantly, this is the place to discuss that.

Because of the wording of the prompt, you might also choose to write about an aspect of your life that isn’t related to ideas of identity. These can include life-altering events, important social interactions you’ve had, or formative experiences.

After you’ve settled on an aspect of identity or an event you deem important enough to write about, consider some questions to help direct your writing:

1) What is the strongest emotion you feel about your chosen aspect of identity or event?

2) Is there a skill, ability, or talent you have due to this aspect or event?

3) Have you developed or strengthened any personality traits as a result of this background? If so, what are they and how have they changed over time?

You don’t necessarily have to include any or all of this information in your response, but if you’re having trouble putting the importance of your chosen aspect/event into words, these questions might inspire some ideas.

Once you’ve chosen a topic and have some idea of how you intend to describe it and its importance, it’s time to write. There are two strong approaches to writing this response:

• The first approach involves doing something totally novel. You might want to pick a completely new aspect of your identity or life event that is fully distinct from one previously mentioned in your application. For example, if you mentioned your gender identity already, you may choose to write about your ethnic background here. This can be a useful approach if certain parts of your identity hold a similar level of importance to you.
• The second approach involves building a previously mentioned attribute/event. Perhaps there is an aspect of your identity that is related to the one you just wrote about, but distinct enough to warrant a new essay. Maybe you talked about being Hispanic somewhere in your application previously, and now want to write about the Spanish language. The language you speak might be an integral part of your identity. It isn’t the same as your ethnic background, even if the two are closely linked, so something like that would be fair game for this prompt.

Though this response is really whatever you want it to be, there are some things you’ll want to try not to do. Remember to avoid simply listing aspects of your identity without elaboration, writing too much about something negative, and discussing a topic that’s too clichéd.

This is another chance to showcase who you are. During the admissions process, there aren’t too many of these opportunities, so make the most of them!

Prompt 6 (Optional)

When not surveying the stars, peering through microscopes, or running through marathons of coding, caltech students pursue an eclectic array of interests that range from speedcubing to participating in varsity athletics to reading romance novels. what is a favorite interest or hobby and why does it bring you joy (100 words).

This prompt is meant to gauge who you are beyond your grades and test scores. It’s an optional prompt, but we strongly recommend writing a response to it, as this gives the admissions committee more knowledge about you.

Caltech wants to know what interests you outside of school, and what hobbies you might bring to their campus. Your hobbies don’t necessarily have to be traditional extracurricular activities, but you might still want to look at our guide to writing the extracurricular activities essay for some tips!

Before you begin writing, it’s important that you select a strong topic. Of course, you need to be sincere. Don’t write about a topic you don’t consider a hobby just because you think the admissions committee wants to read about it. An authentic topic will always make for a better essay than an extravagant one. Make a list of your most meaningful hobbies and consider the following questions:

1) Which hobby on your list have you shown the most commitment to? Which has been most influential in your development?

2) What is the strongest emotion you feel about this hobby?

• Why do you feel this emotion?
• Has that emotional response changed over time? If so, how and why?
• What emotions do you feel during the activity?

3) What thoughts and feelings go through your mind while you participate in this hobby/activity?

4) Have you developed or strengthened any personality traits as a result of this hobby? If so, what are they and how have they evolved over time?

5) Have you developed any skills due to this hobby? These can include soft skills such as critical thinking, public speaking, work ethic, and teamwork, or hard skills, which are specific to whatever domain your hobby is a part of.

6) What impact has this hobby had on the rest of your life (other activities, social life, academics, etc.)?

Once you’ve chosen your hobby, think about how you want to structure your essay. You only have 100 words, which is a very small space to work within, so you’re going to have to be concise. The prompt specifically asks why this interest brings you joy, so you’ll definitely want to include a response to that question.

You have some flexibility in the way you respond to this prompt. You might explicitly state what you enjoy about the hobby, or perhaps you’ll talk about some of the activity’s outcomes that have brought you fulfillment.

Consider this response from a hypothetical student:

“My fingers pluck each string deliberately but delicately. My foot taps quietly along, keeping rhythm like a metronome. I am at peace, once again practicing classical guitar like I have every day for the past ten years. That seems long already, but there is still so much to learn.

As each mellifluous note wafts through the air, I am filled with the joy of knowing there is another technique to master, another piece to play, another obstacle to conquer. Playing classical pieces is more than a hobby; it is a challenge, an opportunity to honor something that transcends time.”

This is a strong response for a number of reasons:

• First, it uses very evocative language to great effect, painting a picture of the hobby in question.
• Second, it describes in detail the emotions the hobby evokes and the reason it elicits joy in the student.
• Finally, it showcases the student’s perspective in a way that cannot be misconstrued. This student is clearly intellectually stimulated by this hobby, dedicated to it, and industrious when it comes to practicing—all excellent qualities to bring to Caltech.

You should strive to do the same things in your essay. Use imagery to your advantage, be specific when discussing your emotions, and try to describe your emotional response to the hobby in a way that reveals something about your personality.

You want to craft an effective essay, so you should note a few common mistakes to avoid:

• Don’t pick the wrong activity! Bad activities include: hobbies you’ve already written about somewhere else in your application, impressive-sounding hobbies you don’t actually participate in, and hobbies you haven’t actually put that much time into.
• Don’t just describe the interest without elaborating on its impact on you. You might get caught up in your anecdote when writing, but don’t forget to explain the hobby’s significance.
• Don’t just list your accomplishments within the hobby. You shouldn’t simply provide a list of things that make you look good superficially. You want to show your personal perspective and growth by discussing your emotional response to your chosen hobby and how the hobby impacts your life.

Structurally, take a reflective approach and really analyze your thoughts and feelings about the hobby. Since you only have 100 words to work with, avoid writing more than one anecdote. You need to be concise in your language, but as long as you can provide a good reflection and describe what it is about your hobby that brings you joy, you will be fine.

Prompt 7 (Optional)

Did you have a hard time narrowing it down to just one interest or hobby we understand – caltech students like to stay busy, too – tell us about another hobby or interest (50 words).

This prompt is also optional, but it’s a great chance to describe something else you’re passionate about. If you were stuck on the previous prompt, struggling to choose between two hobbies that are really important to you, you can describe the second one here.

Bear in mind that this prompt has only 50 words, half the words you had for the previous prompt. If you decide to write a response to this prompt, you have to be extremely precise in your word choice. Consult the guide to the previous prompt above, CollegeVine’s guide to writing the extracurricular activities essay , for more in-depth tips on how you should craft your essay.

Consider the following example response:

“My fingers pluck each string deliberately but delicately. I am at peace, practicing classical guitar like I have daily for the past decade. As notes float through the air, I’m filled with the joy of knowing there is another technique to master, another piece to play, another obstacle to conquer.”

This response is the previous example response adapted to fit the smaller word limit. Notice that it still defines the hobby, paints a picture of the activity, and describes the student’s emotional response when participating in the activity. Of course, the reduced word count means that this essay reveals a bit less about the student than the previous version, but it still answers the prompt well.

Where to Get Your Caltech Essay Edited

Do you want feedback on your Caltech 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

Expository Papers in Mathematics

Introduction.

It is very helpful to have an introduction (and, depending on how long your paper is, an abstract) to contextualize your paper. Introductions should not yet bombard the reader with heavy mathematics. Keep it light. It is often interesting to frame the paper in some interesting question that can only be answered by understanding the material in your paper. Or, if your topic is not something you can tie into a real-world problem, use your introduction to convince the reader of the relevance of the topic. Why is it interesting? What subfields of math is it pertinent to? Historical background, if you choose to use any, finds a great home in the introduction.

As with any paper, it is important to know your audience. If your professor doesn’t specify, ask. Can you assume that the reader has taken your course? Or should your paper be accessible to the general public? With math being in essence its own language, it is important to know how much hand-holding and defining your paper should do. Regardless of your audience, though, you must give concrete examples of the often abstract ideas being discussed.

When in doubt, use examples! Examples are the best way to ground your reader and ensure they understand your complex material. Particularly when your definitions or theorems are in general form, pick a basic yet illustrative example with which to couple it. It is often helpful to give an example before stating the general definition or theorem. That way, the example eases your reader into the general form, as opposed to serving as damage control after having scared them with abstraction.

Let your voice come through! Just because it’s a math paper doesn’t mean it has to uphold the field’s stigma of being dry and clinical-sounding. Your reader should want to read your paper. So, don’t shy away from being punny, taking on a tone, or varying your syntax. And have fun—just as you would in any piece of writing. What is unique about a math paper, though, is that actual definitions, theorems, conjectures, proofs etc. must retain their formalities.

As such, it is helpful to keep your informal prose visually separate from the more precise and technical components of the paper. Paragraphs and punctuation are still crucial in math papers (and the same rules of grammar apply), but the prose certainly can and should be spliced by definitions, examples, theorems, proofs, etc. It would be helpful for the mind and easier on the eye to use boldface and italicized language where appropriate, for distinction and consistency. Additionally, papers are often divided into sections and subsections, in order to make structural sense of the material.

LaTeX is a typesetting language that does much of this formatting for you and is particularly helpful in typing out equations. It is the most commonly used language for writing technical math papers. Some professors may even require you to use LaTeX. If so, Claremont Center for the Mathematical Sciences (CCMS) Software Lab is a helpful resource for learning. They offer tutorials at the library.

Make sure that you understand the conventions of the subfield you’re discussing. Subfields of mathematics have specific ways of expressing things, and the same symbol may mean something totally different in two different subfields. In that vein, clarify what your symbols mean. Now, mathematical shorthand is prevalent in class lectures, but it should not make an appearance in your paper. Just as academic papers refrain from using contractions and abbreviations, your math paper’s prose should use words.

You don’t necessarily need a conclusion, but it is a nice way to wrap up. Many math papers end abruptly, so you really shouldn’t feel obligated to write a conclusion. If you choose to write one, you can bookend it with some of the information relayed in your introduction. It can include more math within it than did the introduction, but it should not leave your reader with the sensation of being overwhelmed. It can also include the limitations or extending applications of the topic/results, and it can highlight open problems and suggest areas for future research.

Bibliography

Lastly, make sure to cite your sources! It is convention to use endnotes to cite information throughout the paper, so it does not interrupted the flow of the sentence. If you are using LaTeX, it compiles a bibliography for you, as you write and cite. Make sure your sources are reliable and valid. Peer reviewed sources are ideal.

For more information on mathematical writing (including tips for writing proofs, posters, and research papers) reference Handbook of Writing for the Mathematical Sciences by Nicholas J. Higham (1998).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

1— Mathematics as a Gateway to Student Success

DALE PARNELL

Oregon State University

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

2— Market Launch

ROL FESSENDEN

L. L. Bean, Inc.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

3— Integrating Vocational and Academic Education

THOMAS BAILEY

Columbia University

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

Rethinking Vocational and Academic Education

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

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

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

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

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

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

Innovative Examples

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

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

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

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

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

Joining Work and Learning

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

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

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

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

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

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

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

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

4— The Importance of Workplace and Everyday Mathematics

JEAN E. TAYLOR

Rutgers University

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

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

What Should Students Learn for Today's World?

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

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

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

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

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

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

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

Workplace and Everyday Settings as a Way of Making Sense

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

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

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

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

Concepts or Procedures?

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

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

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

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

An Example: The Student/Professor Problem

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

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

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

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

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

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

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

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

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

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

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

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

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

5— Working with Algebra

DANIEL CHAZAN

Michigan State University

SANDRA CALLIS BETHELL

Holt High School

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

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

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

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

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

Our Teaching and Issues it Raised

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

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

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

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

An Approach to School Algebra

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

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

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

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

The "Sponsor" Project

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

Using These Questions

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

TABLE 5-1: Questions to ask in the workplace

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

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

Some Examples of Students' Work

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

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

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

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

TABLE 5-2: Cost of carpet worksheet

FIGURE 5-1: Pregnancy wheel

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Emergency Calls

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

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

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

Mathematical Analysis

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

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

FIGURE 1: Distribution of Arrow's response times

FIGURE 2: Distribution of Metro's response times

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

FIGURE 3: Arrow response times by time of day

FIGURE 4: Metro response times by time of day

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

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

Back-of-the-Envelope Estimates

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

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

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

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

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

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

The examples are solved separately below.

Grading Homework

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

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

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

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

The New York Times

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

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

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

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

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

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

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

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

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

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

Scheduling Elevators

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

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

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

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

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

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

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

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

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

Scenario One

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

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

TABLE 1: Elevator round-trip time, Scenario one

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

Scenario Two

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

TABLE 2: Elevator round-trip times, Scenario two

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

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

Scenario Three

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

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

TABLE 3: Elevator round-trip times, Scenario three

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Heating-Degree-Days

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

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

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

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

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

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

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

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

FIGURE 1: Daily heating-degree-days

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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• About the Editors

Math is Like Science, Only with Proof

The slightly unclear photo and shaky handwriting in this diagram brought to you by moving train.

Math is not science.  Sciences seek to understand some aspect of phenomena, and is based on empirical observations, while math seeks to use logic to understand and often prove relationships between quantities and objects which may relate to no real phenomena.  Scientific theories may be supported by evidence, but not proven, while we can actually prove things in math. On the other hand, math is like science, and emphasizing the difference may really work against math.  I find my students often have no sense of how anyone would actually “do math,” for example, how we think of things to try and prove.  Students do seem to have a sense of how people “do science,” and they find it correspondingly less intimidating.  With this in mind I have been working to convince my students that math often works pretty much like science.  Joseph Silverman lays this out in his book A Friendly Introduction to Number Theory (which I am teaching from this semester, and he says this in chapter 1,  accessible here ) really nicely–in number theory, we gather data by computing a lot of examples.  Then we search for a pattern, make a hypothesis, and test it against additional data (more examples).  If the hypothesis doesn’t match the new data, we revise it.  After some iterations, when new data matches our hypothesis, we finally try to prove the hypothesis.

While this step of full logical proof differs from the scientific method, the next part of the process is the same in math as in science: peer review! Science would be a total mess without peer review, likewise math. Simply because it is possible to logically prove things in math, but human fallibility means that our own mistakes may be invisible to us.  Even logic takes a village.

We all know peer review matters—every time we use a published result without having to painstakingly check the proof ourselves, every time we submit a paper which eventually comes back with miraculous reports, from mysterious people who find a host of mistakes, mostly small but occasionally something major (see Adriana Salerno’s post on referees here ).  This has always been a bit amazing to me: at least two anonymous people have taken the time to read and carefully check everything I have ever submitted, without being paid or even identified so I could acknowledge them publicly.  Knowing how long it takes me to read and fully make sense of a paper, this represents a significant outlay of time and energy.  What a public service!  Yay for good referees!

Of course, I’m saying this now because I am myself reviewing a paper for the first time, and it is taking quite a bit of work.  And producing a fair amount of anxiety, I must say. I have been asked to review things before but have not been able to say yes, for various reasons.  This spring I was asked to look at two articles that both looked very interesting and were in areas close to work that I had done.  Shouldn’t be too hard, right?  A few hours each, right?

I am now near the twenty-hour mark on the first article.  Ten of those hours happened on a plane during an epic cross-country flight that was diverted and delayed. At around 1 AM, my red pen started oozing ink after a pressure change and dripped all over the printout I was working with, making it look like my blood, if not necessarily sweat and tears, was literally going into this project. Don’t get me wrong—twenty hours is actually okay, and reviewing a paper is clearly way, way easier than writing one.  This manuscript is well-written, the result is interesting, and I haven’t found any deep issues.  I’ve learned some things by working through it so closely.  Overall this has been really positive; however, I have been at points been racked by terrible doubts.  The early part of the manuscript had some typos and errors in definitions that made many computations impossible to follow.  I was sure that something was wrong, but also sure that I must be missing some obvious reason or solution.  I tried changing the definitions so that the later calculations would hold, but the changes I made seemed to take me in circles.  The authors are mathematicians I really respect and the work is interesting, but my anxiety mounted as I found I couldn’t move forward because I wasn’t sure what the definitions should be.  Eek!  What to do?  Should I send the editor a query to pass on to the author?  Was it okay to ask someone else to look over my work?  I really wanted to do a good job, and I also didn’t want to embarrass myself by making a big deal over something obvious.  Even though I am always encouraging my students to speak up in class, saying there is nothing wrong with asking questions, I really didn’t want to ask what I felt might be a stupid question.

It turned out that after some encouragement from a math friend and a little time away from the article, I found the right small changes to make the paper consistent and was able to move ahead.  All is well.  However, I swore to become a proof-reading fiend in my future writing.  My own errors may be hard for me to see, but they may be even harder for others to fix.  I have a whole new appreciation for referees, and I hope that I can make their lives easier in the future.

This experience also started me wondering about how reviewing/refereeing papers is appreciated or rewarded by the larger mathematical community.  I am happy to do this, and I think it’s important to do this.  I benefit because this made me read an an article I would have wanted to read anyway.  I now know more about my area and that I am just a slightly better mathematician for having worked through this paper carefully.  I have to wonder, though: how much do departments and tenure committees appreciate refereeing as either scholarly or service work?  Do editors take reviewers who write thoughtful reports more seriously?  For a pre-tenure professor, how much referee/review work will benefit a career and how much is too much?  I honestly don’t know any of these things, and I would love some reader feedback on this.

Returning to my starting thoughts: math is not science, but their fates are inextricably linked. We support the communities from within by peer-reviewing, but we have to work together as a larger community to secure support from the outside.  The National Science Foundation’s support for mathematical research has become more and more essential as other funding sources (like the National Security Agency’s Mathematical Science Program ) disappear.  Also, as a lover of logic and believer in the importance of using the power of mathematical thinking to do good in the world, I want to advocate for science-based policy for the common good.  Karen Saxe, director of the Washington office of the AMS, has written some excellent posts about NSF funding for mathematical sciences and even provided a template email to send to your representatives and senators. Though one appropriations request deadline has passed, it is still important and never too late to let them know that you fervently support funding for science and mathematics.  That’s why I’m going to the March for Science this Saturday in Washington DC.  In a lucky or well-planned concomitance, the National Math Festival is also happening on Saturday, about a mile from the site of the march.  I’m headed to both—anyone else?  Maybe we can even lead the march back for some math fun afterward! Sounds like a great weekend for celebrating and supporting math and science. So I’d better get busy and finish my referee report.

Opinions expressed on these pages were the views of the writers and did not necessarily reflect the views and opinions of the American Mathematical Society.

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260 Interesting Math Topics for Essays & Research Papers

Mathematics is the science of numbers and shapes. Writing about it can give you a fresh perspective and help to clarify difficult concepts. You can even use mathematical writing as a tool in problem-solving.

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In this article, you will find plenty of interesting math topics. Besides, you will learn about branches of mathematics that you can choose from. And if the thought of letters and numbers makes your head swim, try our custom writing service . Our professionals will craft a paper for you in no time!

And now, let’s proceed to math essay topics and tips.

🔝 Top 10 Interesting Math Topics

✅ branches of mathematics, ✨ fun math topics.

• 🏫 Math Topics for High School
• 🎓 College Math Topics
• 🤔 Advanced Math
• 📚 Math Research
• ✏️ Math Education
• 💵 Business Math

🔍 References

• Number theory in everyday life.
• Logicist definitions of mathematics.
• Multivariable vs. vector calculus.
• 4 conditions of functional analysis.
• Random variable in probability theory.
• How is math used in cryptography?
• The purpose of homological algebra.
• Concave vs. convex in geometry.
• The philosophical problem of foundations.
• Is numerical analysis useful for machine learning?

What exactly is mathematics ? First and foremost, it is very old. Ancient Greeks and Persians were already utilizing mathematical tools. Nowadays, we consider it an interdisciplinary language.

Biologists, linguists, and sociologists alike use math in their work. And not only that, we all deal with it in our daily lives. For instance, it manifests in the measurement of time. We often need it to calculate how much our groceries cost and how much paint we need to buy to cover a wall.

Simply put, mathematics is a universal instrument for problem-solving. We can divide pure math into three branches: geometry, arithmetic, and algebra. Let’s take a closer look:

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• Geometry By studying geometry, we try to comprehend our physical surroundings. Geometric shapes can be simple, like a triangle. Or, they can form complicated figures, like a rhombicosidodecahedron.
• Arithmetic Arithmetic deals with numbers and simple operations: subtraction, addition, division, and multiplication.
• Algebra Algebra is used when the exact numbers are unclear. Instead, they are replaced with letters. Businesses often need algebra to predict their sales.

It’s true that most high school students don’t like math. However, that doesn’t mean it can’t be a fun and compelling subject. In the following section, you will find plenty of enthralling mathematical topics for your paper.

If you’re struggling to start working on your essay, we have some fun and cool math topics to offer. They will definitely engage you and make the writing process enjoyable. Besides, fun math topics can show everyone that even math can be entertaining or even a bit silly.

• The link between mathematics and art – analyzing the Golden Ratio in Renaissance-era paintings.
• An evaluation of Georg Cantor’s set theory.
• The best approaches to learning math facts and developing number sense.
• Different approaches to probability as explored through analyzing card tricks.
• Chess and checkers – the use of mathematics in recreational activities.
• The five types of math used in computer science.
• Real-life applications of the Pythagorean Theorem .
• A study of the different theories of mathematical logic.
• The use of game theory in social science.
• Mathematical definitions of infinity and how to measure it.
• What is the logic behind unsolvable math problems?
• An explanation of mean, mode, and median using classroom math grades.
• The properties and geometry of a Möbius strip.
• Using truth tables to present the logical validity of a propositional expression.
• The relationship between Pascal’s Triangle and The Binomial Theorem.
• The use of different number types: the history.
• The application of differential geometry in modern architecture.
• A mathematical approach to the solution of a Rubik’s Cube.
• Comparison of predictive and prescriptive statistical analyses.
• Explaining the iterations of the Koch snowflake.
• The importance of limits in calculus.
• Hexagons as the most balanced shape in the universe.
• The emergence of patterns in chaos theory.
• What were Euclid’s contributions to the field of mathematics?
• The difference between universal algebra and abstract algebra.

🏫 Math Essay Topics for High School

When writing a math paper, you want to demonstrate that you understand a concept. It can be helpful if you need to prepare for an exam. Choose a topic from this section and decide what you want to discuss.

• Explain what we need Pythagoras’ theorem for. 
• What is a hyperbola? 
• Describe the difference between algebra and arithmetic. 
• When is it unnecessary to use a calculator ? 
• Find a connection between math and the arts. 
• How do you solve a linear equation? 
• Discuss how to determine the probability of rolling two dice. 
• Is there a link between philosophy and math? 
• What types of math do you use in your everyday life? 
• What is the numerical data? 
• Explain how to use the binomial theorem. 
• What is the distributive property of multiplication? 
• Discuss the major concepts in ancient Egyptian mathematics . 
• Why do so many students dislike math? 
• Should math be required in school? 
• How do you do an equivalent transformation? 
• Why do we need imaginary numbers? 
• How can you calculate the slope of a curve? 
• What is the difference between sine, cosine, and tangent? 
• How do you define the cross product of two vectors? 
• What do we use differential equations for? 
• Investigate how to calculate the mean value. 
• Define linear growth. 
• Give examples of different number types. 
• How can you solve a matrix? 

🎓 College Math Topics for a Paper

Sometimes you need more than just formulas to explain a complex idea. That’s why knowing how to express yourself is crucial. It is especially true for college-level mathematics. Consider the following ideas for your next research project:

• What do we need n-dimensional spaces for?
• Explain how card counting works.
• Discuss the difference between a discrete and a continuous probability distribution .
• How does encryption work?
• Describe extremal problems in discrete geometry.
• What can make a math problem unsolvable?
• Examine the topology of a Möbius strip.

• What is K-theory?  
• Discuss the core problems of computational geometry. 
• Explain the use of set theory . 
• What do we need Boolean functions for? 
• Describe the main topological concepts in modern mathematics. 
• Investigate the properties of a rotation matrix. 
• Analyze the practical applications of game theory.  
• How can you solve a Rubik’s cube mathematically? 
• Explain the math behind the Koch snowflake. 
• Describe the paradox of Gabriel’s Horn. 
• How do fractals form? 
• Find a way to solve Sudoku using math. 
• Why is the Riemann hypothesis still unsolved? 
• Discuss the Millennium Prize Problems. 
• How can you divide complex numbers? 
• Analyze the degrees in polynomial functions. 
• What are the most important concepts in number theory? 
• Compare the different types of statistical methods. 

🤔 Advanced Topics in Math to Write a Paper on

Once you have passed the trials of basic math, you can move on to the advanced section. This area includes topology, combinatorics, logic, and computational mathematics. Check out the list below for enticing topics to write about:

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• What is an abelian group?
• Explain the orbit-stabilizer theorem.
• Discuss what makes the Burnside problem influential.
• What fundamental properties do holomorphic functions have?
• How does Cauchy’s integral theorem lead to Cauchy’s integral formula?
• How do the two Picard theorems relate to each other?
• When is a trigonometric series called a Fourier series?
• Give an example of an algorithm used for machine learning.
• Compare the different types of knapsack problems.
• What is the minimum overlap problem?
• Describe the Bernoulli scheme.
• Give a formal definition of the Chinese restaurant process.
• Discuss the logistic map in relation to chaos.
• What do we need the Feigenbaum constants for?
• Define a difference equation.
• Explain the uses of the Fibonacci sequence.
• What is an oblivious transfer?
• Compare the Riemann and the Ruelle zeta functions.
• How can you use elementary embeddings in model theory?
• Analyze the problem with the wholeness axiom and Kunen’s inconsistency theorem.
• How is Lie algebra used in physics ?
• Define various cases of algebraic cycles.
• Why do we need étale cohomology groups to calculate algebraic curves?
• What does non-Euclidean geometry consist of?
• How can two lines be ultraparallel?

📚 Math Research Topics for a Paper

Choosing the right topic is crucial for a successful research paper in math. It should be hard enough to be compelling, but not exceeding your level of competence. If possible, stick to your area of knowledge. This way your task will become more manageable. Here are some ideas:

• Write about the history of calculus.
• Why are unsolved math problems significant?
• Find reasons for the gender gap in math students.
• What are the toughest mathematical questions asked today?
• Examine the notion of operator spaces.
• How can we design a train schedule for a whole country?
• What makes a number big?

• How can infinities have various sizes?
• What is the best mathematical strategy to win a game of Go?
• Analyze natural occurrences of random walks in biology.
• Explain what kind of mathematics was used in ancient Persia.
• Discuss how the Iwasawa theory relates to modular forms.
• What role do prime numbers play in encryption?
• How did the study of mathematics evolve?
• Investigate the different Tower of Hanoi solutions.
• Research Napier’s bones. How can you use them?
• What is the best mathematical way to find someone who is lost in a maze?
• Examine the Traveling Salesman Problem. Can you find a new strategy?
• Describe how barcodes function.
• Study some real-life examples of chaos theory. How do you define them mathematically?
• Compare the impact of various ground-breaking mathematical equations .
• Research the Seven Bridges of Königsberg. Relate the problem to the city of your choice.
• Discuss Fisher’s fundamental theorem of natural selection.
• How does quantum computing work?
• Pick an unsolved math problem and say what makes it so difficult.

✏️ Math Education Research Topics

For many teachers, the hardest part is to keep the students interested. When it comes to math, it can be especially challenging. It’s crucial to make complicated concepts easy to understand. That’s why we need research on math education.

• Compare traditional methods of teaching math with unconventional ones.
• How can you improve mathematical education in the U.S.?
• Describe ways of encouraging girls to pursue careers in STEM fields.
• Should computer programming be taught in high school?
• Define the goals of mathematics education .
• Research how to make math more accessible to students with learning disabilities .
• At what age should children begin to practice simple equations?
• Investigate the effectiveness of gamification in algebra classes.
• What do students gain from taking part in mathematics competitions?
• What are the benefits of moving away from standardized testing ?
• Describe the causes of “ math anxiety .” How can you overcome it?
• Explain the social and political relevance of mathematics education.
• Define the most significant issues in public school math teaching.
• What is the best way to get children interested in geometry?
• How can students hone their mathematical thinking outside the classroom?
• Discuss the benefits of using technology in math class.
• In what way does culture influence your mathematical education?
• Explore the history of teaching algebra.
• Compare math education in various countries.

• How does dyscalculia affect a student’s daily life?
• Into which school subjects can math be integrated?
• Has a mathematics degree increased in value over the last few years?
• What are the disadvantages of the Common Core Standards?
• What are the advantages of following an integrated curriculum in math?
• Discuss the benefits of Mathcamp.

🧮 Algebra Topics for a Paper

The elegance of algebra stems from its simplicity. It gives us the ability to express complex problems in short equations. The world was changed forever when Einstein wrote down the simple formula E=mc². Now, if your algebra seminar requires you to write a paper, look no further! Here are some brilliant prompts:

• Give an example of an induction proof.
• What are F-algebras used for?
• What are number problems?
• Show the importance of abstract algebraic thinking .
• Investigate the peculiarities of Fermat’s last theorem.
• What are the essentials of Boolean algebra?
• Explore the relationship between algebra and geometry.
• Compare the differences between commutative and noncommutative algebra.
• Why is Brun’s constant relevant?
• How do you factor quadratics?
• Explain Descartes’ Rule of Signs.
• What is the quadratic formula?
• Compare the four types of sequences and define them.
• Explain how partial fractions work.
• What are logarithms used for?
• Describe the Gaussian elimination.
• What does Cramer’s rule state?
• Explore the difference between eigenvectors and eigenvalues.
• Analyze the Gram-Schmidt process in two dimensions.
• Explain what is meant by “range” and “domain” in algebra.
• What can you do with determinants?
• Learn about the origin of the distance formula.
• Find the best way to solve math word problems.
• Compare the relationships between different systems of equations.
• Explore how the Rubik’s cube relates to group theory.

📏 Geometry Topics for a Research Paper

Shapes and space are the two staples of geometry. Since its appearance in ancient times, it has evolved into a major field of study. Geometry’s most recent addition, topology, explores what happens to an object if you stretch, shrink, and fold it. Things can get pretty crazy from here! The following list contains 25 interesting geometry topics:

• What are the Archimedean solids? 
• Find real-life uses for a rhombicosidodecahedron. 
• What is studied in projective geometry? 
• Compare the most common types of transformations. 
• Explain how acute square triangulation works. 
• Discuss the Borromean ring configuration. 
• Investigate the solutions to Buffon’s needle problem. 
• What is unique about right triangles? 

• Describe the notion of Dirac manifolds.
• Compare the various relationships between lines.
• What is the Klein bottle?
• How does geometry translate into other disciplines, such as chemistry and physics?
• Explore Riemannian manifolds in Euclidean space.
• How can you prove the angle bisector theorem?
• Do a research on M.C. Escher’s use of geometry.
• Find applications for the golden ratio .
• Describe the importance of circles.
• Investigate what the ancient Greeks knew about geometry.
• What does congruency mean?
• Study the uses of Euler’s formula.
• How do CT scans relate to geometry?
• Why do we need n-dimensional vectors?
• How can you solve Heesch’s problem?
• What are hypercubes?
• Analyze the use of geometry in Picasso’s paintings.

➗ Calculus Topics to Write a Paper on

You can describe calculus as a more complicated algebra. It’s a study of change over time that provides useful insights into everyday problems. Applied calculus is required in a variety of fields such as sociology, engineering, or business. Consult this list of compelling topics on a calculus paper:

Get an originally-written paper according to your instructions!

• What are the differences between trigonometry, algebra, and calculus?
• Explain the concept of limits.
• Describe the standard formulas needed for derivatives.
• How can you find critical points in a graph?
• Evaluate the application of L’Hôpital’s rule.
• How do you define the area between curves?
• What is the foundation of calculus?

• How does multivariate calculus work?
• Discuss the use of Stokes’ theorem.
• What does Leibniz’s integral rule state?
• What is the Itô stochastic integral?
• Explore the influence of nonstandard analysis on probability theory.
• Research the origins of calculus.
• Who was Maria Gaetana Agnesi?
• Define a continuous function.
• What is the fundamental theorem of calculus?
• How do you calculate the Taylor series of a function?
• Discuss the ways to resolve Runge’s phenomenon.
• Explain the extreme value theorem.
• What do we need predicate calculus for?
• What are linear approximations?
• When does an integral become improper?
• Describe the Ratio and Root Tests.
• How does the method of rings work?
• Where do we apply calculus in real-life situations?

💵 Business Math Topics to Write About

You don’t have to own a company to appreciate business math. Its topics range from credits and loans to insurance, taxes, and investment. Even if you’re not a mathematician, you can use it to handle your finances. Sounds interesting? Then have a look at the following list:

• What are the essential skills needed for business math?
• How do you calculate interest rates?
• Compare business and consumer math.
• What is a discount factor?
• How do you know that an investment is reasonable?
• When does it make sense to pay a loan with another loan?
• Find useful financing techniques that everyone can use.
• How does critical path analysis work?
• Explain how loans work.
• Which areas of work utilize operations research?
• How do businesses use statistics?
• What is the economic lot scheduling problem?
• Compare the uses of different chart types.
• What causes a stock market crash?
• How can you calculate the net present value?
• Explore the history of revenue management.
• When do you use multi-period models?
• Explain the consequences of depreciation.
• Are annuities a good investment?
• Would the U.S. financially benefit from discontinuing the penny?
• What caused the United States housing crash in 2008?
• How do you calculate sales tax?
• Describe the notions of markups and markdowns.
• Investigate the math behind debt amortization.
• What is the difference between a loan and a mortgage?

With all these ideas, you are perfectly equipped for your next math paper. Good luck!

• What Is Calculus?: Southern State Community College
• What Is Mathematics?: Tennessee Tech University
• What Is Geometry?: University of Waterloo
• What Is Algebra?: BBC
• Ten Simple Rules for Mathematical Writing: Ohio State University
• Practical Algebra Lessons: Purplemath
• Topics in Geometry: Massachusetts Institute of Technology
• The Geometry Junkyard: All Topics: Donald Bren School of Information and Computer Sciences
• Calculus I: Lamar University
• Business Math for Financial Management: The Balance Small Business
• What Is Mathematics: Life Science
• What Is Mathematics Education?: University of California, Berkeley
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Classroom Q&A

With larry ferlazzo.

In this EdWeek blog, an experiment in knowledge-gathering, Ferlazzo will address readers’ questions on classroom management, ELL instruction, lesson planning, and other issues facing teachers. Send your questions to [email protected]. Read more from this blog.

Strategies for Using Art in Math, English, Science, and History

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(This is the second post in a three-part series. You can see Part One here. )

The new question-of-the-week is:

What are the best ways you have used art in lessons?

In Part One , Wendi Pillars, Keisha Rembert, Delia M. Cruz-Fernández, and Irina McGrath, Ph.D., offered their suggestions. Wendi, Keisha, and Delia were also guests on my 10-minute BAM! Radio Show. You can also find a list of, and links to, previous shows here.

Today, Sara Rezvi , Gretchen Bernabei, Jeremy Hyler, and Kelsey Pycior share their recommendations.

Art & Math

Sara Rezvi (@arsinoepi on twitter) is a former high school mathematics teacher, a current doctoral candidate at the University of Illinois at Chicago in the College of Education, and the program director of the Math Circles of Chicago, a nonprofit organization that seeks to provide equitable access to rich and thoughtfully designed mathematics outside regular school hours for all children:

Are you a math teacher? How often do you get the following comments when introducing yourself to others?

• Math? Oof, I’m just not a math person.
• Wow, you must be really smart.

As a former high school mathematics teacher and a current doctoral candidate studying mathematics curriculum and instruction, I find these responses deeply saddening. The first suggests that mathematics was something that was done TO the person speaking rather than in community with, and the second response suggests that math continues to remain a proxy for intelligence. Either way, the internalized message I hear from these types of responses is clear: Mathematics is a playground for some, not all.

But what if it’s not mathematics that’s the issue here, but the standard approach in how mathematics has been exceptionalized and taught in the United States?

In our paper, Radical Love as Praxis: Ethnic Studies and Teaching Mathematics for Collective Liberation , my co-authors, Cathery Yeh, Ricardo Martinez, Shraddha Shirude, and I argue how ethnic studies and mathematics engage in reimagining what spaces of mathematics built in community, solidarity, and love might look and feel like.

One of the ethos we describe in our paper is related to the concept of community and solidarity, where we define the following:

“Community and Solidarity in mathematics, as defined by ethnic studies, see mathematics as integral to activist movements for social justice. Ethnic studies is a process that connects learning to the community and to the real world, acknowledging the situated, and body, and Collective nature of learning and change. Mathematics learning is not just experienced cognitively; it is a process that has lived, engraved in students’ bodies and memories, and shaped by our histories, ancestors, and communities.” (Yeh, Martinez, Rezvi, Shirude, 2021, p. 82)

What if mathematics was embodied in the K-12 classroom space? What if it was poetic? What if it was seen as beautiful by our society rather than something to overcome or get through in schooling? What if?

Francis Su ( @mathyawp ) discusses this in the quote below from his book, Mathematics for Human Flourishing :

Photo Credit with permission of Francis Su

What if we celebrated and explored mathematics through art?

In this post, I offer one approach of doing so by exploring Islamic geometry with students that centers the idea of exploration, joy, and solidarity with Muslim students who have and may continue to be experiencing an increasingly hostile climate due to racism, xenophobia, and white supremacy. As educators, it is our collective responsibility to ensure that every child that walks into our classroom space is welcomed and cared for, is seen and heard, and is celebrated in the fullness that they bring. Math class is no exception.

A scroll through the geometry common core standards most math teachers incorporate into their lesson plans reveals language describing students exploring such concepts as symmetry, tessellation, congruency, similarity, and composition. What if art could lend itself in exploring these geometry standards? I describe below an example of this vision through the lens of Islamic geometry.

What do you notice? What do you wonder in the images above? The beautiful tile work you see photographed here is from the AlHambra , the 13 th -century palace and fortress constructed by Narsrid emir Mohammed ben Al-Ahmar. For more insight, John Jaworski’s insightful book, A Mathematician’s Guide to the AlHambra was a rich resource for creating my lesson.

Students were then invited to go on a virtual field trip in small breakout rooms of the 600-year-old Mosque of Sultan Barquq in Cairo. I encourage you to do the same. The link is remarkable, and students enjoyed being able to go on a field trip under COVID lockdown when I first delivered this lesson in the summer of 2020.

What patterns did you see? What geometries are beginning to emerge? Why do some polygons tessellate and others do not? Can we find a way to bring a mathematical lens to the beautiful art we’ve just seen?

Indeed, Maryam Mirzhakhani’s daughter described her mother’s work as painting! The recently published children’s book by Megan Reid and illustrated by Aaliya Jaleel celebrating Mirzhakhani, the first Iranian and woman Fields Medalist, is chockful of ways of understanding how math and art are intertwined.

These questions and more were facilitated during this exploration, where students’ curiosities were piqued by the artwork they had been exploring. The art leads to the mathematics. The mathematics leads to the art. In this way, they are connected. It is a conversation and an invitation to see a subject that has been sterilized by high-stakes testing into something anew.

Annie Perkins ( @anniek_p ) has a gorgeous series of #mathartchallenges that she has curated on her website . The Islamic geometry lesson can be found here: Islamic Geometry Lesson by: Sara Rezvi (@arsinoepi) . For further inspiration and to try out some designs yourself with a simple compass and a ruler, Samira Mian’s Islamic Geometry Art series is thoughtfully designed and intriguing. I hope you can experience the pleasure of exploring how math and art are deeply connected for yourself!

‘Visual Prompts’

Gretchen Bernabei taught English/language arts and reading in Texas for 34 years. You can find her work at www.trailofbreadcrumbs.net :

I was having some trouble getting adolescents to write essays. My department was preparing for some state testing, giving students practice prompts like “write about the importance of friendship” or “write an essay about the importance of honesty.” Many of our students wrote, “Friendship is really important,” or “Honesty is really important,” and not much more.

Clearly, students weren’t plunging into the deeper meanings of those statements; they weren’t grappling with the abstract; they weren’t connecting the thoughts to their world at all. I thought about what those essays were asking students to do. They were asking students to write (and explain) an opinion; to state (and explore) a life lesson; to make (and substantiate) an argument.

And then I tried using artwork along with a “life lesson” statement. Like this:

With an image attached, they suddenly had plenty to say about the statement. At first, I used fine art, paintings I could show on my document camera, and then I moved to photographs. Some experimenting showed us that students like to call the sentences “truisms,” and they could write plenty of thoughts about those truisms. More experimenting taught us that showing the photos without the words was even better: Students liked to gaze at the photo, ask themselves what it’s about, and make up a life lesson that fits. In this case, their truisms evolved from “everyone likes pets” to “sometimes an animal speaks more clearly than a human” or “your dog will never hurt your feelings.” Their truisms became startlingly insightful.

There are so many royalty-free photos available through government agencies that it’s easy to find powerful photos.

The words could be translated into any language; the artwork transcends language.

A student reads the sentence out loud; the teacher asks, “Do you think that’s true?” If they do, they write the sentence at the top of their paper. If they don’t, they revise the sentence so that it’s now true for them.

Some more experimenting helped us develop what we now call the “11-minute essay.” Recalling the cubing exercise made popular by Elizabeth Cowan, we adapted the directions for a timed piece of writing:

Instructions for the 11-Minute Essay

• Write the truism on your paper. For one minute, explain what it means. (Stop them after one minute.)
• Indent and look at your truism. Can you think of a moment from a movie when this sentence was true? Name that movie and tell how this sentence was true in it. You have three minutes. If you can’t think of what to write, just keep looking at the picture.

(Stop them after three minutes.)

• Indent and look at your truism. Can you think of a moment from a book or story when this sentence was true? Name that story and tell how this sentence was true. You have three minutes. If you can’t think of what to write, just keep looking at the picture. (Stop them after three minutes.)
• Indent and look at your truism. How do you personally know it’s true? Tell one moment from your own experience when you saw this was true. If you can’t think of what to write, just keep looking at the picture. (Stop them after three minutes.)
• What does all this leave you wondering about that sentence? Indent and start your last paragraph with the words, “I wonder …” or “This makes me wonder.” Or start it any way you like. You have one minute.

Both students and adults often become surprised at what they produced in such a short amount of time. The directions helped guide and translate onto the paper those thoughts which were already inside the writers. But the photos helped the writers plunge into their own experiences and beliefs.

You can find more visual prompts here and here .

References:

Bernabei, Gretchen and Judi Reimer (2013). Fun-Size Academic Writing for Serious Learning: 101 Lessons and Mentor Texts. Corwin Literacy. Thousand Oaks, CA.

Bernabei, Gretchen (2008). Lightning in a Bottle. Trail of Breadcrumbs Press. San Antonio, Texas.

Cowan, Elizabeth. (1986). Writing. Scott, Foresman. Glenview, Ill.

Art & Science

Jeremy Hyler is a middle school English and media-literacy teacher in Michigan. He has co-authored Create, Compose, Connect! Reading, Writing, and Learning with Digital Tools (Routledge/Eye on Education), From Texting to Teaching: Grammar Instruction in a Digital Age , as well as Ask, Explore, Write . Jeremy blogs at MiddleWeb and hosts his own podcast Middle School Hallways. He can be found on Twitter @jeremybballer and at his website jeremyhyler40.com :

I worked closely with our art teacher on different projects to show cross-curricular connections for the students. While working across the hall from each other, we coordinated two specific activities that went with the science curriculum in my district. First, as my students worked on growing radish plants while experimenting with variables and control, they also were learning about stomata and the process of photosynthesis.

As one of our activities, we would gather different examples of leaves from around our school property. The art teacher would teach the students to do leaf rubbings in their physical-science notebooks by placing the leaf behind the page and rubbing the pencil over the other side of it. It creates a rough sketch of the leaf, and then the students label the rubbings for identification purposes. Later (usually the next day), this would lead to a class discussion on the different species of plants that exist around our school’s property and perhaps why we have such an abundance of certain species. It is an easy cross-curricular activity that allows students to be creative.

In addition to leaf rubbings and working on plant identification, the 8th grade students participate in a project called Salmon in the Classroom. During this project, students raise salmon throughout the school year from an egg to what is called a fry. During this project, students learn about water quality, macroinvertebrates, and other species of fish that exist in our waterways here in Michigan. At the end of the project, students have the opportunity to actively engage with these other species of fish when they release the salmon into a local creek.

One of the extensions of the project is for students to use paint and rubber fish stamps of native species to create an imprint of the fish they chose. The paintings are then hung on a classroom or hallway bulletin board to help younger grade levels get excited about the Salmon in the Classroom Project. It gives them something to look forward to in 8th grade.

Art & History

Kelsey Pycior teaches social studies at Manville High School in central New Jersey:

I teach high school social studies, but art is my hobby. I think that we can utilize art (drawings, paintings, music, poetry, etc.) in our daily lives to better express ourselves and connect with our larger communities. By the time students get to high school, art class is often optional.

I try to incorporate art multiple times per year in my courses. In my current school district, I have encouraged students to express themselves and demonstrate evidence of learning through various art projects that are intertwined with what they are studying. My world history students have hand drawn Instagram posts role-playing as the First, Second, and Third Estates of the French Revolution. Students have gotten on the floor and sketched art upside down, as though they were Michelangelo painting the Sistine Chapel. They showed off their new knowledge of Renaissance art compared with Medieval art in this exercise while gaining appreciation for the talent of painters of that era.

Students have also created murals out of scraps of paper that they wrote Haiku poems on while we were studying Japan. U.S. History 2 students frequently create their own propaganda for topics such as World War I, the Great Depression’s New Deal, and World War II. They’ve created illustrated ABC books about World War II topics and even designed graffiti that they would have painted on the Berlin Wall had they lived in that era. Students also watch performances from the Harlem Renaissance, and we trace it to modern music today.

A favorite activity my U.S. History 2 students participate in is the analysis of Vietnam War protest songs. We listen to music, talk about the lyrics, and connect those to what we see happening with the Vietnam War, the civil rights movement, and other rights activism of the 1960s and 1970s. Without the incorporation of art, history does not come to life.

Thanks to Sara, Gretchen, Jeremy, and Kelsey for contributing their thoughts.

Consider contributing a question to be answered in a future post. You can send one to me at [email protected] . When you send it in, let me know if I can use your real name if it’s selected or if you’d prefer remaining anonymous and have a pseudonym in mind.

You can also contact me on Twitter at @Larryferlazzo .

Education Week has published a collection of posts from this blog, along with new material, in an e-book form. It’s titled Classroom Management Q&As: Expert Strategies for Teaching .

Just a reminder; you can subscribe and receive updates from this blog via email (The RSS feed for this blog, and for all Ed Week articles, has been changed by the new redesign—new ones are not yet available). And if you missed any of the highlights from the first 10 years of this blog, you can see a categorized list below.

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Research in the Mathematical Sciences

Research in the Mathematical Sciences is an international, peer-reviewed journal encompassing the full scope of theoretical and applied mathematics, as well as theoretical computer science.

• Encourages submission of longer articles for more complex and detailed analysis and proofing of theorems.
• Publishes shorter research communications (Letters) covering emerging areas of mathematical research.
• Actively seeks to publish seminal papers in emerging and interdisciplinary areas in all of the mathematical sciences.
• Editors play a pivotal role in soliciting high-quality research papers.
• Editorial decisions are made in a timely manner.

This is a transformative journal , you may have access to funding.

• Thomas Y. Hou,
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Latest articles

Some remarks about \( \rho \) -regularity for real analytic maps.

• Maico Ribeiro
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Higher derivatives of functions with given critical points and values

Matroid products in tropical geometry.

• Nicholas Anderson

Closed orientable surfaces and fold Gauss maps

• C. Mendes de Jesus
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Newton lenses

• Rémi Langevin

Journal updates

Transmission eigenvalues and related spectral problems in scattering theory (submission deadline: june 30, 2021).

This special issue will feature recent developments in the theory and applications of transmission eigenvalues and related spectral problems in direct and inverse scattering theory. The transmission eigenvalue problem is at the heart of inverse scattering theory for inhomogeneous media. It has a deceptively simple formulation but presents a perplexing mathematical structure; in particular it is a non-self-adjoint eigenvalue problem. This subject is rich, active and in the past decade has taken a multitude of directions, including developments in the spectral theory for various operators related to scattering, as well as many applications in inverse scattering problems and imaging. We solicit high quality original research papers targeting results on the theory, computations and applications of these topics.

Guest Editor : Fioralba Cakoni, Rutgers University and Houssem Haddar, CMAP Ecole Polytechnique

Submission Deadline : April 30, 2021

Download full details here:  Transmission eigenvalues and Related Spectral Problems in Scattering Theory (PDF, 19.17 kB)

Mathematical Theory of Machine Learning and Applications (Submission Deadline: 31st August, 2021)

In the past decade, deep learning as a branch of machine learning has influenced scientific computing in a fundamental way. This computational breakthrough presents tremendous opportunities and needs for new perspectives on computational mathematics and related emerging fields, such as approximation theory, operator estimation, numerical PDEs, inverse problems, data-driven modeling of dynamical systems, unsupervised and semi-supervised learnings. This special issue will feature high-quality original research, including (but not limited to) the theoretical and computational developments in these topics. Guest Editors:  John Harlim, Thomas Hou, Jinchao Xu Submission Deadline:  August 31, 2021 Download full details here:  Mathematical Theory of Machine Learning and Applications

PDE Methods for Machine Learning (Submission Deadline: 31st August 2021)

This special issue will feature recent developments in the application of partial differential equations (PDE) to problems in machine learning. We solicit high quality original research papers targeting the analysis and applications of PDEs to problems in machine learning and data science.

Guest Editors:  Jeff Calder (University of Minnesota), Xiuyuan Cheng (Duke University), Adam Oberman (McGill University), Lars Ruthotto (Rutgers University)

Submission Deadline:  31st August 2021

Download Full Details Here: 

PDE Methods for Machine Learning

Developments in Commutative Algebra: In honor of Jürgen Herzog on the occasion of his 80th Birthday

Jürgen Herzog is one of the most accomplished researchers in the modern developments of commutative algebra. He has produced more than 230 original research papers and is cited more than 6650 times by approximately 2150 authors. In honor of his great achievements, we look forward to publishing a special issue commemorating his 80th birthday and honoring his influence on the field of commutative algebra and mathematics in general.

Guest Editor:  Takayuki Hibi  Submission Deadline:  31st December 2021 Download Full Details Here:  Developments in Commutative Algebra

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Mathematics as a Basis in Computer Science Essay

Introduction, correlation between computer science and mathematics, extracurricular activities.

There is a lot of resemblance between mathematics and computer since. Through this essay, I demonstrate how a mastery of mathematics laid a foundation for my interest in computer science and more specifically, website design and computer programming. I have also explored how my love for mathematics helped improve my performance in other subjects as well. This is an indication that mathematics as a discipline cuts across virtually every other subject. Over the years, I have been involved in various extracurricular activities, but none has been as compelling as my desire to help poor children. Again, I demonstrate why I am motivated to assist these needy members of our society.

There is a strong correlation between mathematics and computer science. In this case, the field of computer science borrows heavily from the mathematics discipline. From a very young age, I started manifesting an interest in mathematics. This interest was influence in part by the fact that both of my parents taught mathematics at high school level. Therefore, naturally, the expectation would have been that I would also take an interest in the discipline of my parents, and I did not disappoint them. As a teenager, I discovered that I also had an interest in computer science. Perhaps this was as a result of my good command of such branches of mathematics as discrete mathematics and Boolean algebra. Many will now agree that the involuntary application of computing skills in mathematics is now an indispensable activity. This is a further testament to the connection between the two disciplines. For example, computational thinking or algorithm as used in computer science resembles abstractions development and empirical verifications that finds application in mathematics.

Due to my passion for both computer science and mathematics, I have managed to receive recognition by wining various contests, both in high school and at college level. Winning these contest only served to enhance my resolve to do better in these two disciplines. The two branches, along with several others, find application in the wider field of computer science. Furthermore, computer science requires logical and mathematical applications. These are the necessary ingredients for the design as well as control of computers.

One other reason why mathematics as a discipline fascinates me is because in enables one to solve problems facing them in a very logical manner. In addition, I have also learnt that thanks to my love for mathematics, my thoughts and words process is also precise. Another discovery that I made very early in high school is that mathematics finds application in other disciplines as well. Consequently, my grades in the other subjects were also improving dramatically. For example, my scores in physics and chemistry were also comparable to those I obtained in mathematics, a further testament to the importance of mathematics in other disciplines.

I am really indebted to my high school teacher who discovered my love for mathematics and encouraged me to further my interest in mathematics and computer science. By helping me nurture my skills, I have since managed to retain interest in the two disciplines. My hands –on and practical skills approach has helped shape my desire to pursue computer programming. Over the last 6 years, I have been involved in the designing of websites for clients at a fee. The most interesting thing is that I am a self-taught website designer, an indication of the level to which an individual can go when they are self-motivated. Again, good command for mathematics has come in handy because programming entails a lot for computation in algorithms. Upon completing high school I was involved in the creation of toy projects and small games. This was a necessary move, because I end up borrowing a lot from the experience that I gained then, when I am either undertaking computer programming activities, or in handling large amounts of data in my work.

In today’s business environment, the issue of corporate social responsibility is very common. It is usually perceived as an indication of the fact that an organization would wish to give back to society after their have recorded success in their core activities. However, even individuals are themselves socially responsible. Some prefer to call this action extracurricular activity, while others perceive it as the desire to help those in need. From a personal point of view, I have taken part in numerous extracurricular activities, a majority of which involves the helping of poor children. Some people prefer to spend their spare time with families and friends, sharing in their experiences or just relaxing in a jovial and homely atmosphere. Others prefer to spend their free time in solitude. In my case, this is the time that I dedicate to charity work. I believe the driving force to my participating in charity has been influenced in more ways than one by my mother. She is a social worker by profession. Although her income modest, nonetheless, she manages to dedicate part of her salary to acts of charity, such as footing the hospital bills of friends in need.

Through the years, I have come to appreciate the fact that when you give, one gets more blessings. For example, although my mother is a generous woman, both financially and in terms of advice, we have never lacked the basics in life. She is also at peace with her family friends and more importantly, her creator. I have also personally experienced this during my charity work. I recall giving a homeless person a hundred dollar bill on the eve of one Christmas. Although I had saved this money for a long time, I shall forever remember the bright face of the man to whom I gave that money. This experience taught me the need to share with others the little that I have. A majority of the poor children that I interact with are homeless, and some have no parent. Interacting with these children has also helped me to see the world from their perspective. Given the fact that a majority of the rich have no intention of sharing part of their wealth with the homeless and poor children do they ever wonder what would become of them if these children ganged up against them? Many of these poor children have big dreams that they would wish to fulfill. When I interact with them and they share their fears and dreams with me, I am compelled to do the little I can to help them realize these dreams. This, I believe, is what keeps me motivated to help them.

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IvyPanda. (2024, March 16). Mathematics as a Basis in Computer Science. https://ivypanda.com/essays/mathematics-as-a-basis-in-computer-science/

"Mathematics as a Basis in Computer Science." IvyPanda , 16 Mar. 2024, ivypanda.com/essays/mathematics-as-a-basis-in-computer-science/.

IvyPanda . (2024) 'Mathematics as a Basis in Computer Science'. 16 March.

IvyPanda . 2024. "Mathematics as a Basis in Computer Science." March 16, 2024. https://ivypanda.com/essays/mathematics-as-a-basis-in-computer-science/.

1. IvyPanda . "Mathematics as a Basis in Computer Science." March 16, 2024. https://ivypanda.com/essays/mathematics-as-a-basis-in-computer-science/.

Bibliography

IvyPanda . "Mathematics as a Basis in Computer Science." March 16, 2024. https://ivypanda.com/essays/mathematics-as-a-basis-in-computer-science/.

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Essay on Science for Students and Children

500+ words essay on science.

Essay on science:  As we look back in our ancient times we see so much development in the world. The world is full of gadgets and machinery . Machinery does everything in our surroundings. How did it get possible? How did we become so modern? It was all possible with the help of science. Science has played a major role in the development of our society. Furthermore, Science has made our lives easier and carefree.

Science in our Daily Lives

As I have mentioned earlier Science has got many changes in our lives. First of all, transportation is easier now. With the help of Science it now easier to travel long distances . Moreover, the time of traveling is also reduced. Various high-speed vehicles are available these days. These vehicles have totally changed. The phase of our society. Science upgraded steam engines to electric engines. In earlier times people were traveling with cycles. But now everybody travels on motorcycles and cars. This saves time and effort. And this is all possible with the help of Science.

Secondly, Science made us reach to the moon. But we never stopped there. It also gave us a glance at Mars. This is one of the greatest achievements. This was only possible with Science. These days Scientists make many satellites . Because of which we are using high-speed Internet. These satellites revolve around the earth every day and night. Even without making us aware of it. Science is the backbone of our society. Science gave us so much in our present time. Due to this, the teacher in our schools teaches Science from an early age.

Get the huge list of more than 500 Essay Topics and Ideas

Science as a Subject

In class 1 only a student has Science as a subject. This only tells us about the importance of Science. Science taught us about Our Solar System. The Solar System consists of 9 planets and the Sun. Most Noteworthy was that it also tells us about the origin of our planet. Above all, we cannot deny that Science helps us in shaping our future. But not only it tells us about our future, but it also tells us about our past.

When the student reaches class 6, Science gets divided into three more subcategories. These subcategories were Physics, Chemistry, and Biology. First of all, Physics taught us about the machines. Physics is an interesting subject. It is a logical subject.

Furthermore, the second subject was Chemistry . Chemistry is a subject that deals with an element found inside the earth. Even more, it helps in making various products. Products like medicine and cosmetics etc. result in human benefits.

Last but not least, the subject of Biology . Biology is a subject that teaches us about our Human body. It tells us about its various parts. Furthermore, it even teaches the students about cells. Cells are present in human blood. Science is so advanced that it did let us know even that.

Leading Scientists in the field of Science

Finally, many scientists like Thomas Edison , Sir Isaac Newton were born in this world. They have done great Inventions. Thomas Edison invented the light bulb. If he did not invent that we would stay in dark. Because of this Thomas Edison’s name marks in history.

Another famous Scientist was Sir Isaac Newton . Sir Isaac Newton told us about Gravity. With the help of this, we were able to discover many other theories.

In India Scientists A..P.J Abdul was there. He contributed much towards our space research and defense forces. He made many advanced missiles. These Scientists did great work and we will always remember them.

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