Part One: Connecting Mathematics with Work and Life | High School Mathematics at Work: Essays and Examples for the Education of All Students

Απριλίου 10, 2017

Part One: Connecting Mathematics with Work and Life | High School Mathematics at Work: Essays and Examples for the Education of All Students | The National Academies Press

Part One—
Connecting Mathematics with Work and Life

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Suggested Citation: "Part One: Connecting Mathematics with Work and Life." National Research Council. 1998. High School Mathematics at Work: Essays and Examples for the Education of All Students. Washington, DC: The National Academies Press. doi: 10.17226/5777.

This page in the original is blank.

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Overview

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.

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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-

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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

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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

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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.

References

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.

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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-

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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

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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

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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.

References

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.

Note

1.	For further discussion of these issues, see Parnell (1985, 1995).

DALE PARNELL 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.

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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,

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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

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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

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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 ROI¹ 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. …"

Analysis

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

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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,

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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.

Note

1.	Return on investment.

ROL FESSENDEN 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).

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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

Page 25

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

Page 26

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

Page 27

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.

Page 28

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.

References

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.

Page 29

Notes

Vocational education has been shaped by
federal legislation since the first vocational education act was passed
in 1917. According to the current legislation, the Carl D. Perkins
Vocational and Technical Education Act of 1990, vocational students are
those not headed for a baccalaureate degree, so they include both
students expecting to work immediately after high school as well as
those expecting to go to a community college.

2.	This point of view underlies the reforms articulated in the 1990 reauthorization of the Carl Perkins Vocational and Technical Education Act (VATEA). VATEA also promoted a program, dubbed "tech-prep," that established formal articulations between secondary school and community college curricula.

3.	This argument is reviewed in Bailey & Merritt (1997). For an argument about how education may be organized around broad work themes can enhance learning in mathematics see Hoachlander (1997).

4.

These wage data are reviewed in Levy & Murnane (1992).

5.	The Goals 2000: Educate America Act, for example, established the National Skill Standards Board in 1994 to serve as a catalyst in the development of a voluntary national system of skills standards, assessments, and certifications for business and industry.

THOMAS BAILEY
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.

Page 30

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

Page 31

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?

Page 32

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

Page 33

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, 6S = 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

Page 34

from the diagram, and regarding S and P as units, the student may write 6S = 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 = 6P, 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.

Conclusion

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.

References

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.

JEAN E. TAYLOR 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.

Page 35

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

Page 36

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

Page 37

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

Page 38

TABLE 5-1: Questions to ask in the workplace

QUANTITIES: MEASURED OR COUNTED VERSUS COMPUTED

What quantities are measured or counted by the people you interview?
What kinds of tools are used to measure or count?
Why is it important to measure or count these quantities?
What quantities do they compute or calculate?
What kinds of tools are used to do the computing?
Why is it important to compute these quantities?

COMPUTING QUANTITIES

When a quantity is computed, what information is needed and then what computations are done to get the desired result?
Are there ever different ways to compute the same thing?

REPRESENTING QUANTITIES AND RELATIONSHIPS BETWEEN QUANTITIES

How are quantities kept track of or represented in this line of work?
Collect examples of graphs, charts, tables, etc. that are used in the business.
How is information presented to clients or to others who work in the business?

COMPARISONS

What kinds of comparisons are made with computed quantities?
Why are these comparisons important to do?
What set of actions are set into motion as a result of interpretation of the computations?

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.

Page 39

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

INPUTS			OUTPUT
LENGTH	WIDTH	AREA OF THE ROOM	COST FOR CARPETING ROOM
10	35
20	25
15	30

Page 40

FIGURE 5-1: Pregnancy wheel

Conclusion

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.

References

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.

Page 41

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.

Notes

1.

For other details, see Chazan (1996).
2.

For more detail on high school students' social groups, see Eckert (1989).

3.	Our ideas have been greatly influenced by Schwartz & Yerushalmy (1992) and Yerushalmy & Schwartz (1993) and are in the same spirit as the approach taken by Fey, Heid, et al. (1995), Kieran, Boileau, & Garancon (1996), Nemirovsky (1996), and Thompson (1993).

DANIEL CHAZAN
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.

SANDRA CALLIS BETHELL
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.

Page 42

Emergency Calls

Task

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

DATE OF CALL	TIME OF CALL	COMPANY NAME	RESPONSE TIME IN MINUTES	DATE OF CALL	TIME OF CALL	COMPANY NAME	RESPONSE TIME IN MINUTES
1	7:12 AM	Metro	11	12	8:30 PM	Arrow	8
1	7:43 PM	Metro	11	15	1:03 AM	Metro	12
2	10:02 PM	Arrow	7	15	6:40 AM	Arrow	17
2	12:22 PM	Metro	12	15	3:25 PM	Metro	15
3	5:30 AM	Arrow	17	16	4:15 AM	Metro	7
3	6:18 PM	Arrow	6	16	8:41 AM	Arrow	19
4	6:25 AM	Arrow	16	18	2:39 PM	Arrow	10
5	8:56 PM	Metro	10	18	3:44 PM	Metro	14
6	4:59 PM	Metro	14	19	6:33 AM	Metro	6
7	2:20 AM	Arrow	11	22	7:25 AM	Arrow	17
7	12:41 PM	Arrow	8	22	4:20 PM	Metro	19
7	2:29 PM	Metro	11	24	4:21 PM	Arrow	9
8	8:14 AM	Metro	8	25	8:07 AM	Arrow	15
8	6:23 PM	Metro	16	25	5:02 PM	Arrow	7
9	6:47 AM	Metro	9	26	10:51 AM	Metro	9
9	7:15 AM	Arrow	16	26	5:11 PM	Metro	18
9	6:10 PM	Arrow	8	27	4:16 AM	Arrow	10
10	5:37 PM	Metro	16	29	8:59 AM	Metro	11
10	9:37 PM	Metro	11	30	11:09 AM	Arrow	7
11	10:11 AM	Metro	8	30	9:15 PM	Arrow	8
11	11:45 AM	Metro	10	30	11:15 PM	Metro	8

Page 43

Commentary

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

Page 44

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.

Extensions

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.

Page 45

Back-of-the-Envelope Estimates

Task

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?

Commentary

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 10⁴ dollars. Some corporate
executives and professional athletes make annual salaries on the order
of magnitude of $10,000,000 or 10⁷ dollars. To say that these salaries differ by a factor of 1000 or 10³,
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.

Page 46

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.

Mathematical Analysis

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

Page 47

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 ft³.

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.

Extensions

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?

Page 48

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.

Page 49

Scheduling Elevators

Task

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?

Commentary

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

Page 50

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.

Mathematical Analysis

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

	TIME (SEC)
Ground Floor	25
Floor 1	20
Floor 2	20
Floor 3	20
Floor 4	20
Floor 5	20
Floor 6	20
Return	30
ROUND-TRIP	175

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

Page 51

TABLE 2: Elevator round-trip times, Scenario two

	ELEVATOR A		ELEVATORS B & C
	Stop Time		STOP TIME
Ground Floor		25		25
Floor 1	1	20		5
Floor 2	2	20		5
Floor 3	3	20		5
Floor 4		0	4	20
Floor 5		0	5	20
Floor 6		0	6	20
Return		15		30
ROUND-TRIP		100		130

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

	ELEVATOR A		ELEVATOR B		ELEVATOR C
	STOP TIME		STOP TIME		STOP TIME
Ground Floor		25		25		25
Floor 1	1	20		5		5
Floor 2	2	20		5		5
Floor 3		0	3	20		5
Floor 4		0	4	20		5
Floor 5		0		0	5	20
Floor 6		0		0	6	20
Return		10		20		30
ROUND-TRIP		75		95		115

Page 52

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.

Extensions
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

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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.

Page 54

Heating-Degree-Days

Task

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.

Commentary

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.

Mathematical Analysis

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.

Page 55

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.

Page 56

Extensions

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.