mathematics is the language of science essay

Why Mathematics Is a Language

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Mathematics is called the language of science. Italian astronomer and physicist Galileo Galilei is attributed with the quote, " Mathematics is the language in which God has written the universe ." Most likely this quote is a summary of his statement in  Opere Il Saggiatore:

[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word.

Yet, is mathematics truly a language, like English or Chinese? To answer the question, it helps to know what language is and how the vocabulary and grammar of mathematics are used to construct sentences.

Key Takeaways: Why Math is a Language

• In order to be considered a language, a system of communication must have vocabulary, grammar, syntax, and people who use and understand it.
• Mathematics meets this definition of a language. Linguists who don't consider math a language cite its use as a written rather than spoken form of communication.
• Math is a universal language. The symbols and organization to form equations are the same in every country of the world.

What Is a Language?

There are multiple definitions of " language ." A language may be a system of words or codes used within a discipline. Language may refer to a system of communication using symbols or sounds. Linguist Noam Chomsky defined language as a set of sentences constructed using a finite set of elements. Some linguists believe language should be able to represent events and abstract concepts.

Whichever definition is used, a language contains the following components:

• There must be a vocabulary of words or symbols.
• Meaning must be attached to the words or symbols.
• A language employs grammar , which is a set of rules that outline how vocabulary is used.
• A syntax organizes symbols into linear structures or propositions.
• A narrative or discourse consists of strings of syntactic propositions.
• There must be (or have been) a group of people who use and understand the symbols.

Mathematics meets all of these requirements. The symbols, their meanings, syntax, and grammar are the same throughout the world. Mathematicians, scientists, and others use math to communicate concepts. Mathematics describes itself (a field called meta-mathematics), real-world phenomena, and abstract concepts.

Vocabulary, Grammar, and Syntax in Mathematics

The vocabulary of math draws from many different alphabets and includes symbols unique to math. A mathematical equation may be stated in words to form a sentence that has a noun and a verb, just like a sentence in a spoken language. For example:

3 + 5 = 8

could be stated as "Three added to five equals eight."

Breaking this down, nouns in math include:

• Arabic numerals (0, 5, 123.7)
• Fractions (1⁄4, 5⁄9, 2 1⁄3)
• Variables (a, b, c, x, y, z)
• Expressions (3x, x 2 , 4 + x)
• Diagrams or visual elements (circle, angle, triangle, tensor, matrix)
• Infinity (∞)
• Imaginary numbers (i, -i)
• The speed of light (c)

Verbs include symbols including:

• Equalities or inequalities (=, <, >)
• Actions such as addition, subtraction, multiplication, and division (+, -, x or *, ÷ or /)
• Other operations (sin, cos, tan, sec)

If you try to perform a sentence diagram on a mathematical sentence, you'll find infinitives, conjunctions, adjectives, etc. As in other languages, the role played by a symbol depends on its context.

International Rules

Mathematics grammar and syntax, like vocabulary, are international. No matter what country you're from or what language you speak, the structure of the mathematical language is the same.

• Formulas are read from left to right.
• The Latin alphabet is used for parameters and variables. To some extent, the Greek alphabet is also used. Integers are usually drawn from i , j , k , l , m , n . Real numbers are represented by  a ,  b ,  c , α , β , γ. Complex numbers are indicated by w and z . Unknowns are x , y , z . Names of functions are usually f , g , h .
• The Greek alphabet is used to represent specific concepts. For example, λ is used to indicate wavelength and ρ means density.
• Parentheses and brackets indicate the order in which the symbols interact .
• The way functions, integrals, and derivatives are phrased is uniform.

Language as a Teaching Tool

Understanding how mathematical sentences work is helpful when teaching or learning math. Students often find numbers and symbols intimidating, so putting an equation into a familiar language makes the subject more approachable. Basically, it's like translating a foreign language into a known one.

While students typically dislike word problems, extracting the nouns, verbs, and modifiers from a spoken/written language and translating them into a mathematical equation is a valuable skill to have. Word problems improve comprehension and increase problem-solving skills.

Because mathematics is the same all over the world, math can act as a universal language. A phrase or formula has the same meaning, regardless of another language that accompanies it. In this way, math helps people learn and communicate, even if other communication barriers exist.

The Argument Against Math as a Language

Not everyone agrees that mathematics is a language. Some definitions of "language" describe it as a spoken form of communication. Mathematics is a written form of communication. While it may be easy to read a simple addition statement aloud (e.g., 1 + 1 = 2), it's much harder to read other equations aloud (e.g., Maxwell's equations). Also, the spoken statements would be rendered in the speaker's native language, not a universal tongue.

However, sign language would also be disqualified based on this criterion. Most linguists accept sign language as a true language. There are a handful of dead languages that no one alive knows how to pronounce or even read anymore.

A strong case for mathematics as a language is that modern elementary-high school curricula uses techniques from language education for teaching mathematics. Educational psychologist Paul Riccomini and colleagues wrote that students learning mathematics require "a robust vocabulary knowledge base; flexibility; fluency and proficiency with numbers, symbols, words, and diagrams; and comprehension skills."

• Ford, Alan, and F. David Peat. " The Role of Language in Science ." Foundations of Physics 18.12 (1988): 1233–42. 
• Galilei, Galileo. "'The Assayer' ('Il Saggiatore' in Italian) (Rome, 1623)." The Controversy on the Comets of 1618 . Eds. Drake, Stillman and C. D. O'Malley. Philadelphia: University of Pennsylvania Press, 1960. 
• Klima, Edward S., and Ursula Bellugi. "The Signs of Language. "Cambridge, MA: Harvard University Press, 1979. 
• Riccomini, Paul J., et al. " The Language of Mathematics: The Importance of Teaching and Learning Mathematical Vocabulary ." Reading & Writing Quarterly 31.3 (2015): 235-52. Print.
• Popular Math Terms and Definitions
• Definition and Examples of Syntax
• What Is Lexicogrammar?
• Observations on What Is Language
• 10 Types of Grammar (and Counting)
• Where Did Language Come From? (Theories)
• Definition and Examples of Speakers in Language Studies
• English Grammar: Discussions, Definitions, and Examples
• Universal Grammar (UG)
• Lexis Definition and Examples
• Definition and Examples of Text in Language Studies
• What is Lexicology?
• Definition and Usage of Union in Mathematics
• Cognitive Grammar
• Generative Grammar: Definition and Examples
• 8 Infinity Facts That Will Blow Your Mind

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Physics > Popular Physics

Title: mathematics : the language of science.

Abstract: The purpose of this essay is to bring out the unique role of Mathematics in providing a base to the diverse sciences which conform to its rigid structure. Of these the physical and economic sciences are so intimately linked with mathematics, that they have become almost a part of its structure under the generic title of Applied Mathematics. But with the progress of time, more and more branches of Science are getting quantified and coming under its ambit. And once a branch of science gets articulated into a mathematical structure, the process goes beyond mere classification and arrangement, and becomes eligible as a candidate for enjoying its predictive powers! Indeed it is this single property of Mathematics which gives it the capacity to predict the nature of evolution in time of the said branch of science. This has been well verified in the domain of physical sciences, but now even biological sciences are slowly feeling its strength, and the list is expanding.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

1— Mathematics as a Gateway to Student Success

DALE PARNELL

Oregon State University

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

2— Market Launch

ROL FESSENDEN

L. L. Bean, Inc.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

3— Integrating Vocational and Academic Education

THOMAS BAILEY

Columbia University

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

Rethinking Vocational and Academic Education

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

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

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

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

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

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

Innovative Examples

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

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

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

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

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

Joining Work and Learning

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

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

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

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

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

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

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

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

4— The Importance of Workplace and Everyday Mathematics

JEAN E. TAYLOR

Rutgers University

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

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

What Should Students Learn for Today's World?

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

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

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

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

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

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

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

Workplace and Everyday Settings as a Way of Making Sense

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

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

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

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

Concepts or Procedures?

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

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

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

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

An Example: The Student/Professor Problem

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

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

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

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

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

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

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

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

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

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

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

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

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

5— Working with Algebra

DANIEL CHAZAN

Michigan State University

SANDRA CALLIS BETHELL

Holt High School

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

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

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

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

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

Our Teaching and Issues it Raised

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

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

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

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

An Approach to School Algebra

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

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

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

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

The "Sponsor" Project

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

Using These Questions

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

TABLE 5-1: Questions to ask in the workplace

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

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

Some Examples of Students' Work

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

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

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

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

TABLE 5-2: Cost of carpet worksheet

FIGURE 5-1: Pregnancy wheel

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Emergency Calls

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

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

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

Mathematical Analysis

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

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

FIGURE 1: Distribution of Arrow's response times

FIGURE 2: Distribution of Metro's response times

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

FIGURE 3: Arrow response times by time of day

FIGURE 4: Metro response times by time of day

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

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

Back-of-the-Envelope Estimates

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

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

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

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

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

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

The examples are solved separately below.

Grading Homework

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

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

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

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

The New York Times

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

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

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

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

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

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

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

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

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

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

Scheduling Elevators

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

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

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

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

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

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

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

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

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

Scenario One

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

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

TABLE 1: Elevator round-trip time, Scenario one

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

Scenario Two

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

TABLE 2: Elevator round-trip times, Scenario two

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

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

Scenario Three

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

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

TABLE 3: Elevator round-trip times, Scenario three

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Heating-Degree-Days

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

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

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

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

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

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

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

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

FIGURE 1: Daily heating-degree-days

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Universe Today

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Mathematics: The Beautiful Language of the Universe

Let us discuss the very nature of the cosmos. What you may find in this discussion is not what you expect. Going into a conversation about the universe as a whole, you would imagine a story full of wondrous events such as stellar collapse, galactic collisions, strange occurrences with particles, and even cataclysmic eruptions of energy. You may be expecting a story stretching the breadth of time as we understand it, starting from the Big Bang and landing you here, your eyes soaking in the photons being emitted from your screen. Of course, the story is grand. But there is an additional side to this amazing assortment of events that oftentimes is overlooked; that is until you truly attempt to understand what is going on. Behind all of those fantastic realizations, there is a mechanism at work that allows for us to discover all that you enjoy learning about. That mechanism is mathematics, and without it the universe would still be shrouded in darkness. In this article, I will attempt to persuade you that math isn’t some arbitrary and sometimes pointless mental task that society makes it out to be, and instead show you that it is a language we use to communicate with the stars.

We are currently bound to our solar system. This statement is actually better than it sounds, as being bound to our solar system is one major step up from being bound simply to our planet, as we were

before some very important minds elected to turn their geniuses toward the heavens. Before those like Galileo, who aimed his spyglass towards the sky, or Kepler discovering that planets move about the sun in ellipses, or Newton discovering a gravitational constant, mathematics was somewhat  limited, and our understanding of the universe rather ignorant. At its core, mathematics allows a species bound to its solar system to probe the depths of the cosmos from behind a desk. Now, in order to appreciate the wonder that is mathematics, we must first step back and briefly look at its beginnings and how it is integrally tied into our very existence.

Mathematics almost certainly came about from very early human tribes (predating Babylonian culture which is attributed to some of the first organized mathematics in recorded history), that may have used math as a way of keeping track of lunar or solar cycles, and keeping count of animals, food and/or people by leaders. It is as natural as when you are a young child and you can see that you have

one toy plus one other toy, meaning you have more than one toy. As you get older, you develop the ability to see that 1+1=2, and thus simple arithmetic seems to be interwoven into our very nature. Those that profess that they don’t have a mind for math are sadly mistaken because just as we all have a mind for breathing, or blinking, we all have this innate ability to understand arithmetic. Mathematics is both a natural occurrence and a human designed system. It would appear that nature grants us this ability to recognize patterns in the form of arithmetic, and then we systematically construct more complex mathematical systems that aren’t obvious in nature but let us further communicate with nature.

All this aside, mathematics developed alongside of human development, and carried on similarly with each culture that was developing it simultaneously. It’s a wonderful observation to see that cultures that had no contact with one another were developing similar mathematical constructs without conversing. However, it wasn’t until mankind decidedly turned their mathematical wonder towards the sky that math truly began to develop in an astonishing way. It is by no mere coincidence that our scientific revolution was spurred by the development of more advanced mathematics built not to tally sheep or people, but rather to further our understandings of our place within the universe. Once Galileo began measuring the rates at which objects fell in an attempt to show mathematically that the mass of an object had little to do with the speed in which it fell, mankind’s future would forever be altered.

This is where the cosmic perspective ties in to our want to further our mathematical knowledge. If it were not for math, we would still think we were on one of a few planets orbiting a star amidst the backdrop of seemingly motionless lights. This is a rather bleak outlook today compared to what we now know

about the awesomely large universe we reside in. This idea of the universe motivating us to understand more about mathematics can be inscribed in how Johannes Kepler used what he observed the planets doing, and then applied mathematics to it to develop a fairly accurate model (and method for predicting planetary motion) of the solar system. This is one of many demonstrations that illustrate the importance of mathematics within our history, especially within astronomy and physics.

The story of mathematics becomes even more amazing as we push forward to one of the most advanced thinkers humanity has ever known. Sir Isaac Newton, when pondering the motions of Halley’s Comet, came to the realization that the math that had been used thus far to describe physical motion of massive

bodies, simply would not suffice if we were to ever understand anything beyond that of our seemingly limited celestial nook. In a show of pure brilliance that lends validity to my earlier statement about how we can take what we naturally have and then construct a more complex system upon it, Newton developed the Calculus in which this way of approaching moving bodies, he was able to accurately model the motion of not only Halley’s comet, but also any other heavenly body that moved across the sky.

In one instant, our entire universe opened up before us, unlocking almost unlimited abilities for us to converse with the cosmos as never before. Newton also expanded upon what Kepler started. Newton recognized that Kepler’s mathematical equation for planetary motion, Kepler’s 3rd Law ( P 2 =A 3 ), was purely based on empirical observation, and was only meant to measure what we observed within our solar system. Newton’s mathematical brilliance was in realizing that this basic equation could be made universal by applying a gravitational constant to the equation, in which gave birth to perhaps one of the most important equations to ever be derived by mankind; Newton’s Version of Kepler’s Third Law.

What Newton realized was that when things move in non-linear ways, using basic Algebra would not produce the correct answer. Herein lays one of the main differences between Algebra and Calculus. Algebra allows one to find the slope (rate of change) of straight lines (constant rate of change), whereas Calculus allows one to find the slope of curved lines (variable rate of change). There are obviously many more applications of Calculus than just this, but I am merely illustrating a fundamental difference between the two in order to show you just how revolutionary this new concept was. All at once, the motions of planets and other objects that orbit the sun became more accurately measurable, and thus we gained the ability to understand the universe a little deeper. Referring back to Netwon’s Version of Kepler’s Third Law, we were now able to apply (and still do) this incredible physics equation to almost anything that is orbiting something else. From this equation, we can determine the mass of either of the objects, the distance apart they are from each other, the force of gravity that is exerted between the two, and other physical qualities built from these simple calculations.

With his understanding of mathematics, Newton was able to derive the aforementioned gravitational constant for all objects in the universe ( G = 6.672×10 -11 N m 2 kg -2 ). This constant allowed him to unify astronomy and physics which then permitted predictions about how things moved in the universe. We could now measure the masses of planets (and the sun) more accurately, simply according to Newtonian physics (aptly named to honor just how important Newton was within physics and mathematics). We could now apply this newfound language to the cosmos, and begin coercing it to divulge its secrets. This was a defining moment for humanity, in that all of those things that prohibited our understandings prior to this new form of math were now at our fingertips, ready to be discovered. This is the brilliance of understanding Calculus, in that you are speaking the language of the stars.

There perhaps is no better illustration of the power that mathematics awarded us then in the discovery of the planet Neptune. Up until its discovery in September of 1846, planets were discovered simply by observing certain “stars” that were moving against the backdrop of all the other stars in odd ways. The term planet is Greek for “wanderer”, in that these peculiar stars wandered across the sky in noticeable patterns at different times of the year. Once the telescope was first turned upwards towards the sky by Galileo, these wanderers resolved into other worlds that appeared to be like ours. If fact, some of these worlds appeared to be little solar systems themselves, as Galileo discovered when he began recording the moons of Jupiter as they orbited around it.

After Newton presented his physics equations to the world, mathematicians were ready and excited to begin applying them to what we had been keeping track of for years. It was as if we were thirsty for the knowledge, and finally someone turned on the faucet. We began measuring the motions of the planets and gaining more accurate models for how they behaved. We used these equations to approximate the mass of the Sun. We were able to make remarkable predictions that were validated time and again simply by observation. What we were doing was unprecedented, as we were using mathematics to make almost impossible to know predictions that you would think we could never make without actually going to these planets, and then using actual observation to prove the math correct. However, what we also did was begin to figure out some odd discrepancies with certain things. Uranus, for instance, was behaving not as it should according to Newton’s laws.

What makes the discovery of Neptune so wonderful was the manner in which it was discovered. What Newton had done was uncover a deeper language of the cosmos, in which the universe was able to reveal more to us. And this is exactly what happened when we applied this language to the orbit of Uranus. The manner in which Uranus orbited was curious and did not fit what it should have if it was the only planet that far out from the sun. Looking at the numbers, there had to be something else out there perturbing its orbit. Now, before Newton’s mathematical insights and laws, we would have had no reason to suspect anything was wrong in what we observed. Uranus orbited in the way Uranus orbited; it was just how it was. But, again revisiting that notion of mathematics being an ever increasing dialogue with the universe, once we asked the question in the right format, we realized that there really must be something else beyond what we couldn’t see. This is the beauty of mathematics writ large; an ongoing conversation with the universe in which more than we may expect is revealed.

It came to a French mathematician Urbain Le Verrier who sat down and painstakingly worked through the mathematical equations of the orbit of Uranus. What he was doing was using Newton’s mathematical equations backwards, realizing that there must be an object out there beyond the orbit of Uranus that was also orbiting the sun,

and then looking to apply the right mass and distance that this unseen object required for perturbing the orbit of Uranus in the way we were observing it was. This was phenomenal, as we were using parchment and ink to find a planet that nobody had ever actually observed. What he found was that an object, soon to be Neptune, had to be orbiting at a specific distance from the sun, with the specific mass that would cause the irregularities in the orbital path of Uranus. Confident of his mathematical calculations, he took his numbers to the New Berlin Observatory, where the astronomer Johann Gottfried Galle looked exactly where Verrier’s calculations told him to look, and there lay the 8th and final planet of our solar system, less than 1 degree off from where Verrier’s calculations said for him to look. What had just happened was an incredible confirmation of Newton’s gravitational theory and proved that his mathematics were correct.

These types of mathematical insights continued on long after Newton. Eventually, we began to learn much more about the universe with the advent of better technology (brought about by advances in mathematics). As we moved into the 20th century, quantum theory began to take shape, and we soon realized that Newtonian physics and mathematics seemed to hold no sway over what we observed on the quantum level. In another momentous event in human history, yet again brought forth by the advancement in mathematics, Albert Einstein unveiled his theories of General and Special Relativity, which was a new way to look not only at gravity, but

also on energy and the universe in general. What Einstein’s mathematics did was allow for us to yet again uncover an even deeper dialogue with the universe, in which we began to understand its origins.

Continuing this trend of advancing our understandings, what we have realized is that now there are two sects of physics that do not entirely align. Newtonian or “classical” physics, that works extraordinarily well with the very large (motions of planets, galaxies, etc…) and quantum physics that explains the extremely small (the interactions of sub-atomic particles, light, etc…). Currently, these two areas of physics are not in alignment, much like two different dialects of a language. They are similar and they both work, but they are not easily reconcilable with one another. One of the greatest challenges we face today is attempting to create a mathematical grand “theory of everything” which either unites the laws in the quantum world with that of the macroscopic world, or to work to explain everything solely in terms of quantum mechanics. This is no easy task, but we are striving forward nonetheless.

As you can see, mathematics is more than just a set of vague equations and complex rules that you are required to memorize. Mathematics is the language of the universe, and in learning this language, you are opening yourself up the core mechanisms by which the cosmos operates. It is the same as traveling to a new land, and slowly picking up on the native language so that you may begin to learn from them. This mathematical endeavor is what allows us, a species bound to our solar system, to explore the depths of the universe. As of now, there simply is no way for us to travel to the center of our galaxy and observe the supermassive black hole there to visually confirm its existence. There is no way for us to venture out into a Dark Nebula and watch in real time a star being born. Yet, through mathematics, we are able to understand how these things exist and work. When you set about to learn math, you are not only expanding your mind, but you are connecting with the universe on a fundamental level. You can, from your desk, explore the awesome physics at the event horizon of a black hole, or bear witness to the destructive fury behind a supernova. All of those things that I mentioned at the beginning of this article come into focus through mathematics. The grand story of the universe is written in mathematics, and our ability to translate those numbers into the events that we all love to learn about is nothing short of amazing. So remember, when you are presented with the opportunity to learn math, accept every bit of it because math connects us to the stars.

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18 Replies to “Mathematics: The Beautiful Language of the Universe”

I certainly agree that mathematics can be used to accurately describe many observations. However, there has been a colossal misuse of math today. It has been used to create reality instead of describe it. The best examples are the mathematical creation of dark matter and black holes. Neither of these concepts has ever been observed and they defy confirmation. They are merely mathematical constructs. What brought about the need for these constructs? The fact that astrophysicists and cosmologists have ruled that gravity must be the driving force for galaxy formation. When it was determined that gravity had insufficent forces to cause galaxy formation and maintain galactic structure the need for other forces (unseen and unknown, of course) was apparent. And oila, we have dark matter and black holes. Even with these add-ons intended to salvage gravity-based theories there are still inadequate forces. The use of math to salvage failed theories has led the field of cosmology down a dead end road. Only now with the newest technology and advances in radio telescopes do we see the real forces behind star and galaxy formation. Massive magnetic fields are now detected surrounding “black holes” which are now more correctly being referred to as galactic centers. It is becoming more likely that these centers are super dense plasmoids. There are galactic sized magnetic fields surrounding galaxies as well. Intergalactic electric currents (“rivers of hydrogen”) connecting stars and galaxies have been identified and it is likely that these currents are associated with equally vast magnetic fields. At Los Alamos National Lab, Anthony Pratt used plasma, magnetic fields and electric current to demonstrate spiral galaxy formation in his plasma physics lab. AND these lab induced “spiral galaxies” had the necessary velocities to maintain their shape. Black holes and dark matter were not required. These observations could be and were mathematically described. It seems that we may be leaving the age of Newton and entering the age of Faraday and Tesla. Progress can now be made in cosmology towards rational explanations of observations without the need for imaginary mathematical constructs. The use of the failed gravity based theories to explain the newest observations has resulted in “mysteries” and contradictions. When observed with consideration given to electromagnetic principles the mysteries disappear. I think the presentation by Donald Scott to NASA at the Goddard Colloquia on Engineering in 2009 is a good starting point for one to review the contributions that can be made to the field of cosmology by electrical engineers and plasma physicists. There are many that he references in this presentation that have done ground breaking work in the fields of electromagnetism and plasma physics that can be directly applied to the latest observations from radio telescopes. This work can clearly be used to explain “confounding mysteries” that exist in the field of cosmology. Without knowledge of a “Z pinch”, astrophysicists are left guessing about routine electrical events. Math can then be used to confirm these observations and electromagnetic based explanations. Science fiction can then be left to the novelists.

No. It’s not.

Nope, the electric universe still does not exist.

I recommend public lectures at Canadian Institute for Theoretical Astrophysics which are available on youtube here: http://www.youtube.com/user/citaseminars They have more than a dozen talks about astronomical magnetic fields. I think it has been a neglected topic because it has been hard to observe. But nowadays it is possible to map large scale magnetic field lines by observing the polarization of photons (something like that). But I never heard anyone of them talk about how that could somehow replace gravity or dark matter or affecting the rotation of galaxies. There is indeed an “electric universe” of sorts, but it does not conflict with established physics. Dark matter is carefully mapped. Black holes are observed.

Me and… me and my very, very small but tight-knit internet community knows more… more about outer space stuff than all the outer space science guys out there and… and… and I’m just a-gonna keep… keep a-coming back here and letting you all know over and over and over and over again because it shows the whole wide WORLD how much more smarter I am than… than all of you guys put toGETHer! ….Ya…! You’ll see…. You’ll ALL see someday. You WATCH!

Typical misconception. Dark matter was not just conjured up by some mathematician one day as a whim. It was postulated as a way of explaining observations, and only after all other hypotheses failed for one reason after another. That includes all weird appeals to magnetic or electric fields.

Scientists do not use mathematics to “create reality”, but to describe what they see.

I appreciate the awesome article. Thank you for your military also.

I second that.

Very clear and informing article. Congratulations to its author. But I cannot agree with readers comments based on the “Electric Universe” theory. That theory has no serious base and is not supported by correct math, not even by a thorough understanding of electromagnetic theory. Not to mention that the existence of singularities has nothing to do with the gravitational stability of galaxies.

Oh, they don’t care about being wrong. They just feel that they need something for themselves to believe in.

“Mathematics is the language of the universe”

This is actually not true.

The universe has a completely different language that mathematics. But mathematics is generic enough to be uses to describe and predict simplified models (e.g. 4 forces) of this universe. Mathematics can also be used to describe pure fictional worlds like used in gaming.

It is up to experiments to determine what mathematical formulas really exists in this universe and which are fictional.

But math is just logic. Physical reality is logical. Math looks generic because math in and of itself uses numbers and anonymous variables like “x”, precisely for the purpose of allowing maximum degrees of freedom. But once you use measured physical entities and use them mathematically, then math reveals what is true about the physical reality beyond what was measured. Even if you logically use physically impossible symbols, like negative numbers, during the calculation.

Great Article! And no buts about it.

This is an excellent article!

Galileo wrote, “La mathematca è l’ alfabeto nel quale DIO ha scritto l’ universo.” “Mathematics is the alphabet in which God wrote the universe. The alphabet, not the language.

Some scientists marvel at how mathematical the universe seems to be. But the nature of the universe does not necessarily flow from the mathematics. Mathematics is very flexible and the mathematics is chosen to fit the observations. Once a good fit is found it is then tested to see if it makes valid predictions. We must not assume the underlying nature of the universe is mathematical. Mathematics is just very powerful in representing it. The randomness of some quantum mechanical events may be the point where mathematics fails to mirror the real nature of the universe.

Maybe math might not be the problem but the user?

Reality is Reality twist it any way you want if it makes you happy but that does Not make it Real 🙂 Amen

Nice article, but once again as previously stated, without physically observing something in its true mult dimension universe we live in,the mathematical principles used to describe our cosmos are at the reins of the observer, thus allowing personal beliefs and persuasion. Truth be told that just because one mathematical model fits one answer doesnt mean that theres only one answer but rather there can still be multi answers of movement for that particular mathematical model. For example how and why do we teach and continue to believe facts that mathematical models used to describe movements in our galaxy and solar system in a 2 dimensional platform when we know we live in a multi dimension universe and galaxy. Just cause those mathematical principles fit those porpotions doesnt exclude milti answers in fact its much easier to fit a complex mathematical principle in a multi dimension into a simple 2 dimensional model than it is to do the opposite, which brings us to the reality that jyst like history, change is a constant and to solely rely on one apparatus like math to hold as absolute truth is just as ignorant as believing that the world is flat just cause its the simplest answer. We need to stop believing that theres only one way. Our current knowledge of the cosmos is still very immature and new discoveries everday are changing past theories that mathematical principles used worked but now have to be changed. As long as people want be self rightous and ego centric and so fixated on past theories thought to be true and excepted and take offense to new ways, then we are just Like the Roman Church that drove us into the dark ages and blocked our current knowledge from existed. Please people who consist of our scientific astronomy fans, stop repeating the mistakes of the past and stop being so fixated on your own views as right and CHANGE your mind to be accepting to change because thats the only way truths will be unveiled and we can progressive forward. I think its sad that a website named universetoday can be so self rightous on their views and be willingly to accept only mathematical principles and models as absolute truths kniwing that history as always proven this wrong.

Comments are closed.

Language of Physics, Language of Math: Disciplinary Culture and Dynamic Epistemology

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• Published: 14 March 2015
• Volume 24 , pages 561–590, ( 2015 )

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• Edward F. Redish 1 &
• Eric Kuo 2  

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Mathematics is a critical part of much scientific research. Physics in particular weaves math extensively into its instruction beginning in high school. Despite much research on the learning of both physics and math, the problem of how to effectively include math in physics in a way that reaches most students remains unsolved. In this paper, we suggest that a fundamental issue has received insufficient exploration: the fact that in science, we don’t just use math, we make meaning with it in a different way than mathematicians do. In this reflective essay, we explore math as a language and consider the language of math in physics through the lens of cognitive linguistics. We begin by offering a number of examples that show how the use of math in physics differs from the use of math as typically found in math classes. We then explore basic concepts in cognitive semantics to show how humans make meaning with language in general. The critical elements are the roles of embodied cognition and interpretation in context. Then, we show how a theoretical framework commonly used in physics education research, resources, is coherent with and extends the ideas of cognitive semantics by connecting embodiment to phenomenological primitives and contextual interpretation to the dynamics of meaning-making with conceptual resources, epistemological resources, and affect. We present these ideas with illustrative case studies of students working on physics problems with math and demonstrate the dynamical nature of student reasoning with math in physics. We conclude with some thoughts about the implications for instruction.

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1 Introduction/Problem Statement

Mathematics is deeply woven into both the teaching and practice of physics. Footnote 1 Other sciences, such as chemistry, biology, geology, and meteorology often use math extensively in practice, but typically rely less heavily on it than physics does in high school and college instruction. Despite decades of experience in teaching physics with math from the high school to the graduate level, and despite years of research on “problem solving” in physics (Maloney 1994 ; Hsu et al. 2004 ), the physics community continues to have mixed success in teaching students to use math effectively in physics. A few students seem to take naturally and comfortably to the use of math to describe the physical world, but many struggle with it, both at the introductory and at the more advanced level, even though multiple math classes may be required as prerequisites for physics classes. For example, a physics major at the University of Maryland is required to take five semesters of mathematics and those intending to go on to graduate school are encouraged to take at least two more.

One of us (EFR) has been a faculty member in a large research university physics department for more than 40 years. In this time, he has observed or participated in four major re-considerations of the physics major’s curriculum. In each of these, a critical element of the discussion has been the question, “Why do so many students seem unable to use math in our physics classes, despite having had success in their prerequisite math classes?” This question is raised at all levels, from the introductory (“They can’t do simple algebra!”), to the upper division (“They have terrible trouble with simple matrix concepts!”), to the graduate (“Many of them can’t take a simple Fourier transform!”).

Sometimes, the problem is that students have failed to learn what they have been taught in their math class, and they struggle with basic mathematical concepts and manipulations. But from both our research and teaching experience, we suspect that often the issue is subtler and often has to do with students having difficulty in translating the way symbology is used to make meaning differently in math and science classes. In this reflective essay, in order to see how this might play out, we consider the issue of the use of mathematics in science from a semantic point of view, using the tools of linguistics to talk about how meaning is made.

First, we argue that there are dramatic, if often unrecognized, differences in how the disciplinary cultures of mathematics and physics use and interpret mathematical expressions. In Sect.  2 , we demonstrate through examples how the interpretation of math by physics instructors can blend in physical knowledge, changing the interpretation of the math. The result is that the equations that students learn in their math classes may look and feel very different from similar equations that they see in their science classes (especially physics). Some of these differences are deep.

The key point is this:

Although the formal mathematical syntax may be the same across the disciplines of mathematics and physics, the uses and meanings of that formal syntax may differ dramatically between the two disciplines. These differences in meaning may be masked by a similarity in the formal syntax.

The question we are raising is “How do the different disciplines make meaning with symbols?” But this implicitly relies on the answer to the question “What does meaning mean?” Since this is a subtle and difficult issue, in Sect.  3 we reach out to the disciplines of linguistics and semantics to explore the fundamental processes associated with meaning-making in the use of everyday language and bridge these processes of meaning-making to the use of mathematics.

Next, we seek to understand how students manage these multiple languages across the disciplines. In Sect.  4 we turn to making meaning with math in physics by briefly summarizing the resources theoretical framework of student cognition. This is based on a knowledge-in-pieces model modulated by dynamic framing through student expectations, epistemology, and affect. Interestingly, this structure fits well with what we have drawn from the more general research in linguistics and semantics. Finally, we consider in detail a few case studies of students reasoning with equations in physics, and we show how knowledge, epistemology, and affect interact dynamically to make different kinds of meaning for students. Section  5 gives a brief discussion of the implications for instruction.

2 Math is Different in a Physics Context

Math in science (and particularly math in physics) is not the same as doing math. It has a different purpose—representing meaning about physical systems rather than expressing abstract relationships. It even has a distinct semiotics—the way meaning is put into symbols—from pure mathematics.

It almost seems that the “language” of mathematics we use in physics is not the same as the one taught by mathematicians. There are many important differences in what seems to be the physicist’s “dialect” of speaking math, so, while related, the languages of “math in math” and “math in physics” may need to be considered as separate languages. The key difference is that loading physical meaning onto symbols does work for physicists and leads to differences in how physicists and mathematicians interpret equations. We not only use math in doing physics , we use physics in doing math . We present three examples illustrating different aspects of the cultural differences between the use of math by physicists and mathematicians and then discuss the general structure of mapping meaning to math.

2.1 Loading Meaning Onto Symbols Leads to Differences in How Physicists and Mathematicians Interpret Equations

2.1.1 corinne’s shibboleth.

Our first example is the amusing and surprising parable of “Corinne’s Shibboleth.” Footnote 2 (Dray and Manogoue 2002 ) The particular example shown in Fig.  1 tends to separate physicists from mathematicians. Try it for yourself before reading the discussion that follows.

A problem whose answer tends to distinguish mathematicians from physicists

The context of the problem encourages you to think in terms of a particular physical system. Physicists tend to think of T as a physical function —one that represents the temperature (in whatever units) at a particular point in space (in whatever coordinates). Mathematicians tend to consider T as a mathematical function —one that represents a particular functional dependence relating a value to a pair of given numbers. Footnote 3

As a result, physicists tend to answer that \( T(r,\theta ) = kr^{2} \) because they interpret \( x^{2} + y^{2} \) physically as the square of the distance from the origin. If r and θ are the polar coordinates corresponding to the rectangular coordinates x and y , the physicists’ answer yields the same value for the temperature at the same physical point in both representations. In other words, physicists assign meaning to the variables x , y , r , and θ —the geometry of the physical situation relating the variables to one another.

Mathematicians, on the other hand, may regard x , y , r , and θ as dummy variables denoting two arbitrary independent variables. The variables ( r , θ ) or ( x , y ) do not have any meaning constraining their relationship. Mathematicians focus on the mathematical grammar of the expression rather than any possible physical meaning. The function as defined instructs one to square the two independent variables, add them, and multiply the result by k. The result should therefore be \( T\left( {r,\theta } \right) \, = k(r^{2} + \theta) \) .

Typically, a physicist will be upset at the mathematician’s result. You might hear, “You can’t add r 2 and θ ! They have different units!” The mathematician is likely to be upset at the physicist’s result. You might hear, “You can’t change the functional dependence without changing the name of the symbol!” You have to write something like

To which the physicist might respond, “You can’t write that the temperature equals the entropy! That will be too confusing.” (Physicists often use S to represent entropy). Footnote 4

Note that this is a parable, not an experiment. We are exaggerating the roles of the mathematician and physicist to illustrate the point that one could have different interpretations of the same expression. In reality, practicing physicists and mathematicians lie somewhere in between these two extremes. There are times when physicists may pay careful attention to functional form—for example, when considering the transformation from a Lagrangian to a Hamiltonian or between thermodynamic potentials. Similarly, a mathematician’s back-of-the-envelope work may give preference to quick, intuitive meaning rather than formal syntax. And mathematicians are perfectly capable of reading and interpreting T ( r , θ ) as an informal polar transformation. Yet, we have given this problem dozens of times and seen the described results. Where we have crossovers, the mathematicians who give the “physicist’s answer” tend to be applied mathematics researchers, and the physicists who give the “mathematician’s answer” tend to be abstract mathematical physicists. (And we have often seen non-math major students in an introductory physics class give the mathematician’s answer.)

In their disciplinary context, the mathematicians are right, and the physicists appear to be treading a dangerous path. Once we put numbers in, the expression T (10, 30) is ambiguous. Are we supposed to use rectangular or polar coordinates to evaluate that expression? For a physicist working with measurements that have units, there is no problem since specifying “10 cm and 30°” would identify the expression as being in polar coordinates, but a mathematician in an introductory class who wants to “keep the mathematics clean” might prefer to work with pure numbers. Then, their response is obviously justified.

The fact that physicists “load” physical meaning onto both symbols and numbers in a way that mathematicians usually do not is both powerful and useful. It allows physicists to work with complex mathematical quantities without introducing the fancy math that would be required to handle some issues with mathematical rigor.

2.1.2 Filtering an Equation Through the Physics Changes How We Interpret It

A second way that physicists blend physical meaning into math is through “filtering the equation through the physics.” The way an equation is used can be strongly affected by the physics it is meant to describe. A nice example is the equation for the photoelectric effect.

This example comes from the modern physics section of the calculus-based physics class for engineers. In the photoelectric effect, electrons bound in a metal by an energy of at least Φ absorb photons of energy hf where f is the frequency of the light. If the energy (frequency) of the light is high enough, electrons will be knocked out of the metal. As usual, the instructor (EFR) tried to “twist” student expectations a bit by making small but important modifications to standard problems. The problem shown in Fig.  3 was a surprise to some of the students, but the results were more of a surprise to the instructor.

This one is pretty easy if you keep the physics to the fore. A longer wavelength corresponds to a smaller frequency so the photons have less energy after the modification. If the photons did not have enough energy to knock out an electron originally, they would certainly not have enough if their energy is reduced even more. Yet, nearly a quarter of the class responded that after the modification, there would be electrons knocked out. Their reasoning relied on the Einstein photoelectric effect equation (Fig.  2 ):

A problem where the equation hides some physics

where e is the charge on the electron, V 0 is the stopping potential, f is the frequency of the photon, and Φ is the work function of the metal. One student reasoned as follows: “If hf – Φ is zero before the change, then it will not be zero after the change. Therefore, since it’s not zero, there will be electrons knocked out.”

This reasoning highlights the fact that Eq. ( 2 ) is not actually the correct equation. There is a hidden Heaviside (theta) function that corresponds to the statement: “Do not use this equation unless the right-hand side, which corresponds to the maximum kinetic energy of the knocked out electron, is a positive number.” A mathematical equation that includes this constraint is

Typically, expert physicists (and introductory physics texts) do not bother to be explicit about this theta function. Instead of encoding the conditions of use in the mathematical formalism, physicists apply their understanding of the physical situation—in this case, checking that the energy of the photon is sufficient to knock out electrons—to evaluate if and how this equation should be used.

2.1.3 Interpreting Symbols Physically Yields Hidden Functions

Because their interest is in mathematical relationships, mathematicians teaching math classes focus on the mathematical grammar of an equation, ignoring possible physical meaning. Novice students often do the same in their physics class, to their confusion. A few years ago, one of us (EFR) gave the example shown in Fig.  3 as a quiz question to his second-semester class in algebra-based physics.

A quiz problem that students often misinterpret

The instructor had discussed the definition of electric field extensively, and he had made it “absolutely clear” (including proving the results mathematically) that the strength of the electric field was independent of the test charge—making the correct answer (b). Yet on this quiz, more than half of nearly 200 students thought that the answer should be (c). Upon discussing it with the class after the quiz, it became clear that, although many of them could quote the result “the electric field is independent of the test charge,” most of the students answering incorrectly had not thought to access that knowledge. They looked at the equation \( E = F/q \) and treated the problem as a problem in mathematical grammar. If \( E = F/q \) , then \( F/\left( {{-}3q} \right) = {-}E/3 \) .

In this situation, however, F is not an arbitrary fixed symbol. It represents the Coulomb force that the test charge feels as a result of its interaction with the source charges. It thus is not independent of the strength of the test charge. It increases proportionally as the test charge is increased, resulting in the same electric field as before.

The instructor might have written the force on the test charge as F q or F ( q ) in order to remind the students that the force in fact explicitly depended on the test charge (and after this experience, he sometimes did so). But the culture of physics expects that each symbol in an equation is to be interpreted in conjunction with its physical meaning. So, part of the acculturation of a physics student is learning to interpret the math physically, not to only focus on mathematical structure and manipulations.

2.2 Modeling the Physical World Mathematically

Using mathematical expressions in the disciplines of math or physics is a complex task. In the culture of math, this complexity arises from reasoning and operating in a well-defined and coherent mathematical structure with a particular formal syntax. Yet, the examples in Sect.  2.1 clearly demonstrate that the use of equations in physics goes beyond interpreting and processing the formal mathematical syntax. Instead of relying on explicit Heaviside functions or functional dependences, physicists’ use of math is often informed by “checking the physics.”

More precisely, in the culture of physics, the use of mathematical expressions is complex, because the ancillary physical meaning of symbols is used to convey information omitted from the mathematical structure of the equation. This is because we have a different purpose for the math: to model real physical systems.

2.2.1 Physicists and Mathematicians have Different Goals for the Use of Math

It is not just that physicists read and use equations differently from the way mathematicians do in math classes. Their goals are different. Physicists do not just want to explore the mathematical formalisms, they want to leverage those formalisms to describe, learn about, and understand physical systems.

In order to explicate the various components of the modeling process for the purpose of thinking about teaching mathematical physics, we use the diagram shown in Fig.  4 . (Redish 2005 ; Redish and Smith 2008 )

A model of mathematical modeling

We begin our description of the modeling process in the lower left by deciding what we are doing. We choose some aspects of a particular physical system we want to describe. We identify measurables—variables and parameters that we can quantify through some process of measurement (at least in principle, if not always in practice). We then decide what particular mathematical structures are appropriate for describing the features we are interested in. We then model the physical system by mapping these measurements into mathematical symbols and expressing the physical–causal relations between the measured quantities in terms of mathematical operations between the symbols.

From the mathematical structures we have chosen, we inherit transformational rules and methodologies for transforming relationships and solving equations. We can then process the math to solve problems, leading to answers we were unable to see directly from our physical understanding. This, however, leaves us with only mathematics.

We then have to interpret what the result means back in the physical system. Finally, we have to evaluate whether the result supports our original choice of model when compared to observations or whether it indicates a need to modify our model.

Each of these four steps—modeling, processing, interpreting, and evaluating—are critical skills in the toolbox of a scientist who uses math to describe the behavior of the physical world. This diagram helps a teacher focus on more than just the mathematical processing that often tends to dominate instruction in physics.

But the use of math in physics is not so simple or sequential as this diagram may tend to indicate. The physics and the math get intimately tangled, as we have seen in the examples above. For many professional physicists, the evaluation part—the part showing where the mathematical mapping does not work—is the most important one. Correcting partial models and inventing new ones are the place where physics is at its most creative. But the figure serves to emphasize that our traditional way of thinking about using math in physics classes may not give enough emphasis to the critical elements of modeling, interpreting, and evaluating.

Physics instruction tends to provide our students with ready-made often over-simplified (toy) models, and we may be exasperated—or even irritated—if students focus on details that we know to be irrelevant to the toy model we are considering. We tend to let them do the mathematical manipulations in the process step, but we rarely ask them to interpret their results and even less frequently ask them to evaluate whether the initial model is adequate. At the introductory level, our exams often require only one-step recognition, giving “cues” so we do not require our students to recognize deep structures. When they do not succeed on their own with complex problem solving, we tend to pander by only giving simple problems, on which success is not evidence of problem-solving expertise. We often do not recognize what is complex in a problem for a student and that makes it hard to design appropriate and effective problems.

Our examples show that physicists (as well as other scientists and engineers) often use ancillary physical knowledge—often implicit, tacit, or unstated—when applying mathematics to physical systems. Interestingly enough, a similar idea is valuable to linguists trying to understand how we put meaning to words—semantics. Developments in the linguistics and semantics literature help us to begin to build a terminology to be able to better describe the difference between the expectations of the cultures of physicists and mathematicians.

3 What Do We Mean by Making Meaning? Cognitive Semantics and Mathematics

To understand how we make meaning using the language of mathematics in the context of physics, let us consider what is known about how people make meaning using language in the context of daily life. Cognitive linguists have made considerable progress on this issue in the past 40 years. Although researchers have not entirely come to an overarching synthesis, they have many ideas that are valuable in helping us make sense of how we make meaning. We offer an exceedingly brief summary of a rich and complex subject, selecting those elements that are particularly relevant. We draw heavily on the work of Lakoff and Johnson ( 1980 ), Lakoff and Nuñez ( 2001 ), Langacker ( 1987 ), Fauconnier ( 1985 , 1997 ), Turner ( 1991 ), Fauconnier and Turner ( 2003 ), Evans ( 2006 ), and the overview of Evans and Green ( 2006 ).

From their textbook on modern cognitive semantics, Evans and Green identify core ideas through which meaning is made. We draw on three of these core ideas here (Evans and Green 2006 , p. 157):

Embodied cognition: Meaning is grounded in physical experience.

Encyclopedic knowledge : Ancillary knowledge creates meaning.

Contextualization : Meaning is constructed dynamically.

In this section, we will elaborate on how these principles specify how meaning is made in language and then show how those same mechanisms can be used to understand how meaning is made with mathematical expressions in both science and math.

3.1 Embodied Cognition: Meaning is Grounded in Physical Experience

Embodied cognition refers to the interaction of complex cognitive functions with basic physics experience—our sensory perceptions, motor functions, and how they are tied to cultural contexts. Infants learn what a shape is by coordinating their vision and touch. Toddlers learn the names of these shapes by associating words they hear to objects.

There are abundant examples in everyday language and conceptualization where meaning requires relating to our bodily existence in the three-dimensional world that we experience. Lakoff and Johnson ( 1980/2003 ) discuss extensive examples of many spatial orientation concepts and metaphors such as “up,” “down,” “front,” and “back” that are tied closely to our spatial experiences:

… Thus UP is not understood purely in its own terms but emerges from the collection of constantly performed motor functions having to do with our erect position relative to the gravitational field that we live in. (Lakoff & Johnson, 2004, p. 56 - 57).

This then forms the basis of structuring and understanding more abstract concepts. Metaphorical statements such as: “I’m feeling up ” or “He is really low these days” are conceptualized on the basis of physical orientations.

Lakoff and Johnson ( 1980/2003 ) point out how our bodily experiences with physical objects form the basis of conceptualizing emotions such as anger, love, and events such as inflation in terms of discrete entities or physical substances. Such conceptualization forms the basis on which we use and understand statements such as, “Inflation is backing us into a corner,” and “You’ve got too much hostility in you.” Many of these are so ingrained and automatic that we do not even realize the metaphorical nature of them in everyday use.

The thesis of embodied cognition states that ultimately our conceptual system is grounded in our interaction with the physical world: How we construe even highly abstract meaning is constrained by and is often derived from our very concrete experiences in the physical world. Note that embodied cognition is not a reference to the cognitive activity that is obviously involved in performing sensorimotor activities. The idea is that (a) our close sensorimotor interactions with the external world strongly influence the structure and development of higher cognitive facilities, and (b) the cognitive routines involved in performing basic physical actions are involved in even in higher-order abstract reasoning (Füster 1999 ).

The grounding of conceptualization in physical experience and actions also extends to higher cognitive processes such as mathematical reasoning. Lakoff and Nuñez ( 2001 ) argue that our understanding of many mathematical concepts relies on everyday ideas such as spatial orientations, groupings, bodily motion, physical containment, and object manipulations (such as rotating and stretching). Although formal mathematics may, at times, seem distinct from everyday life, one’s cognition about and understanding of such formal rules may be grounded in common, physical experiences. Thus, an understanding of the mathematics of set theory can be seen as grounded in our physical experience with containers and collections of objects, and an understanding of the arithmetic of complex numbers “makes use of the everyday concept of rotation.” (p. 29). Many of the sophisticated ideas and formulations in mathematics are intricately entwined with the physicality of our being. Footnote 5 , Footnote 6

For example, one way that “embodiment” allows mathematics to feel as if it has meaning, even in abstraction, is through symbolic forms (Sherin 1996 , 2001 , 2006 ). A symbolic form blends a grammatical signifier—a mathematical symbol template —with an abstraction of an understanding of relationships obtained from embodied experience—a conceptual schema .

One example of a symbolic form is parts - of - a - whole . A symbol template for parts-of-a-whole looks like:

The boxes indicate that any symbol may be in the box—we are only talking about the grammatical structure of the mathematical representation. The conceptual meaning put to it is that something can be considered to be made up of parts. This is something with which we have both direct physical experience and an abstract schema of parts and whole. The symbolic form is considered to be the blend of the symbol template with the conceptual schema: The boxes on the right-hand side of the equation each take on the conceptual meaning of “part,” and the box on the left-hand side takes on the conceptual meaning of “whole,” which consists of the sum of all the parts.

Another example of a symbolic form is base  +  change. The symbol template for base  +  change is written as:

The conceptual structure is that the final amount (box on left-hand side) is equal to the starting amount (box on the right-hand side) plus how much that starting amount changes by (triangle). Again, this conceptual schema is something with which we have direct physical experience and where we can expect to have abstracted a schema of change.

3.2 Encyclopedic Knowledge: Ancillary Knowledge Creates Meaning

The principle of encyclopedic knowledge implies that we understand the meaning of words not in terms of terse definitions provided in a dictionary but in reference to a contextual web of concepts (perhaps represented by other words) that are themselves understood on the basis of still other concepts.

Take the example of the word “hypotenuse,” defined as the longest side of a right-angled triangle. To understand the meaning of “hypotenuse,” one needs to understand the meaning of a triangle, “sides” of a triangle, right angle, and the idea of “longest.” These in turn require conceptualizing a plane, shapes on a plane, lines, angles, length, and so on. Understanding and conceptualizing “hypotenuse” relies on a number of ancillary concepts, which in turn rely on a background of other concepts in an expanding web of encyclopedic knowledge.

This idea of encyclopedic knowledge has been developed under various frameworks such as frame semantics (Fillmore 1976 ), domains (Langacker 1987 ), and mental spaces (Turner 1991 ). The idea common to these varied perspectives is that we can model knowledge as consisting of a large number of highly interconnected elements. At any given moment, only part of the network is active, depending on the present context and the history of that particular network. The meaning of a word in a particular utterance in a particular context is determined by what part of that complex web of knowledge is accessed by that particular utterance. Modern cognitive linguists argue compellingly for these complex links in the structure and processing of meaning:

…a number of scholars…have presented persuasive arguments for the view that words in human language are never represented independently of context. Instead, these linguists argue that words are always understood with respect to frames or domains of experience….so that the ‘meaning’ associated with a particular word (or grammatical construction) cannot be understood independently of the frame with which it is associated. (Evans and Green 2006 , p. 211)

At first look, the meaning of mathematical equations could seem terse in the dictionary sense. What does it mean to know the meaning of an equation? Consider for example:

In the strictest sense, it is a statement suggesting a simple calculation, defining the value of the quantity “ y ” in terms of the sum the product of “ m ” and “ x ” and “ b ”. Mathematics sometimes intentionally adopts such a minimalist view. What you know about a mathematical quantity is specified as precisely and as minimally as possible (with axioms), and only that knowledge is to be used in constructing new knowledge. Footnote 7

Though written mathematics can be terse and precise, the culture of math often relies on more than just a “dictionary meaning” in how a symbolic string is used and understood. For most mathematicians (and even high school students), Eq. ( 6 ) carries more meaning than the literal relation between four algebraic and two relational symbols. With a knowledge of labeling conventions, x and y are interpreted as variables capable of taking on many different values, while m and b are interpreted as constants. With this addition, the equation takes on the meaning of a relation between the independent variable ( x ) and the dependent variable ( y ). Additionally, the assumed constancy of m implies that the equation refers to a straight line. The constants now take on additional mathematical meaning: m as the slope of the line and b as the intercept on the y- axis, bringing in ideas from graphs. Thus, the meaning of the equation, understood even within the domains of mathematical conventions, straight lines, and graphs are much richer then the strict “definition” expressing the symbol “ y ” in terms of other symbols.

When we use math in physics, our rich knowledge about the physical system is additionally brought to bear in interpreting the mathematical meaning as illustrated by the examples in Sect.  2 . As another example, think about the equation from physics

where v is the velocity of an object, v 0 is the initial velocity, a is the constant acceleration of the object, and t is the time. This equation is mathematically identical to Eq. ( 6 ) and may be seen as representing a graph of a linear relationship. Footnote 8 But in the context of physics, it is connected to an even richer network of ideas involving motion, speeds, and rates of change. To understand this equation is to understand its relation to all these stores of knowledge. In other words, the meaning of equations in physics is encyclopedic just like the meaning of a word such as “hypotenuse” is encyclopedic. We will see a specific example of how students can perceive this equation in either a mathematical or physical context in Sect.  4.2.1 .

3.3 Contextualization: Meaning is Constructed Dynamically

One implication of the idea that meaning is built by linking to elements of an encyclopedic knowledge system is that the meaning of entities is not fixed but is dynamically determined based on specific contextual clues. Semanticists see linguistic units as prompts for the construction of meaning, or contextualization. Footnote 9 As described by Evans and Green this is:

a dynamic process whereby linguistic units serve as prompts for an array of conceptual operations and the recruitment of background knowledge. It follows from this view that meaning is a process rather than a discrete ‘thing’ that can be ‘packaged’ by language. Meaning construction draws upon encyclopaedic knowledge…and involves inferencing strategies that relate to different aspects of conceptual structure, organization and packaging… [Emphasis in original.] (Evans and Green 2006 , p. 162)

If utterances provide access to nodes in the network of knowledge, different parts of which are active in different moments, then the meaning of an utterance depends on the local activity of the network at any given moment. It is in this sense that the meaning is dynamic. The contexts of a particular utterance—local (what is the conversation about, with whom, etc.) and global (personal histories, social and institutional affordances, etc.)—affect the activity of the knowledge network and, in turn, the contextualization at any given moment. Thus, the meaning of an utterance is not predetermined but is constructed in the moment.

Consider the example of the word “safe” in the following sentences, both referring to a child playing on the beach (Evans and Green 2006 , p. 161):

The child is safe.

The beach is safe.

The first sentence refers to the child as free from harm. The second sentence, though identical in structure to the first, does not imply that the beach is free from harm. Rather, it implies that the beach cannot cause harm. The senses of “safe” in the two sentences (and the properties they assign to child and beach) are slightly different depending on the local context. In a different context, for example, if referring to a beach threatened by property developers, the contextualization and the meaning constructed for the second utterance could be more similar to the first meaning of safe—meaning that the beach is free from being harmed by the developers.

In light of this, we can revisit the example of the imaginary physicist and mathematician arguing about whether T ( r , θ ) should be “ \( r^{2} + \theta^{2} \) ” or “ \( r^{2} \) .” Although both could likely understand the thinking behind the two responses, each expert makes meaning according to the meanings common to their respective disciplinary cultures and the context they bring to it. We can now understand that the mathematicians who ignore the variable names are attaching meaning to the equation within the domain of simple functions and variables, while the physicists who ignore the implications of functional dependence are interpreting the equation within the domain of coordinate systems adding physical meaning to the variables.

Similarly, the idea of symbolic forms does not suggest a 1-to-1 correspondence between a particular equation and a symbol template or between a symbol template and a conceptual schema. Just as the meaning of the word “safe” can depend on the context of use, the same mathematical symbol template can, in different contexts, support different conceptual meanings. For example, \( v = v_{0} + at \) may commonly be understood as base  +  change, whereas an energy equation with the same structure, \( E = \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \, mv^{2} + {mgh} \) , may commonly be interpreted as parts - of - a - whole , that the total mechanical energy consists of the sum of two parts: the kinetic energy and the potential energy. Yet, it may also at times be productive to think about this same energy equation as base  +  change (or change  +  base ), say if a baseball is initially being held at the top of a tall building and is then given kinetic energy by being thrown horizontally.

Making of meaning with equations shares (at least) three key commonalities with meaning-making in language: an embodied basis, the use of encyclopedic knowledge, and contextual selection of that encyclopedic knowledge for meaning-making.

4 Making Meaning in Physics: The Resources Framework

The ideas in Sect.  3 tie in nicely with the resources framework being developed by the Physics Education Research (PER) community for analyzing student thinking in physics (diSessa 1993 ; Hammer 2000 ; Hammer et al. 2005 ; Redish 2004 ; Redish 2014 ). Footnote 10 In this framework, an individual’s reasoning is modeled as resulting from the activation of a subset of cognitive resources (tightly bound cognitive knowledge structures), activated in response to a perception and interpretation of both external and internal contexts (framing).

This framework, used in PER to understand the dynamics of how students construct meaning in physics, shares many of the key features of frameworks used to understand the dynamics of how individuals construct meaning in language.

Embodied cognition: Phenomenological primitives tie basic physics reasoning to embodied experience.

Encyclopedic knowledge : Manifold productive resources are used dynamically.

Contextualization : Activation depends on conceptual, epistemological, and affective factors.

Here, we provide examples of how the resources framework uses these same core ideas to help us describe the multiple ways in which physics students can make meaning with equations, as well as understand the dynamics of how students can shift from one meaning to another and how conceptual, epistemological, and affective responses can interact in complex ways.

4.1 Embodied Cognition: Phenomenological Primitives Tie Basic Physics Reasoning to Embodied Experience

The resources framework puts its “feet on the ground” in a manner similar to cognitive linguistics—through embodied experience. diSessa ( 1993 ) identified basic embodied elements of physics thinking as phenomenological primitives (“p-prims”). These are knowledge elements learned, often at a very young age, about how the world works. Two of their core aspects are obviousness and irreducibility —p-prims are activated easily and directly, and, as far as the user is aware, they have no structure. “That’s just the way things are.” Two examples [from many cited in diSessa’s classic ( 1993 ) paper] are “dynamic balancing” (two influences in conflict which happen to balance each other) and “Ohm’s p-prim” (an agent or impetus acts through a resistance to produce a result). For example, in thinking about how much a heavy object will move in response to a push, a student may draw on their physical intuition through Ohm’s p-prim rather than formal physics principles in their reasoning.

In a similar way, Sherin ( 2001 ) found that upper-division physics undergraduates commonly construct novel equations to model physical situations through their intuitive understanding rather than the application of formal physics rules or principles. In describing the forces on an object falling at terminal velocity, a pair of students skipped directly to writing an equation of the form \( F_{\text{gravity}} = F_{{{\text{air}}\,{\text{resistance}}}} \) . This one-step derivation precludes formal derivation from Newton’s laws, where the total force is found to be \( F_{\text{net}} = F_{\text{gravity}} - F_{\text{air\,resistance}} \) , that is inserted into Newton’s second law to give \( F_{\text{net}} = F_{\text{gravity}} - F_{\text{air\,resistance}} = ma \) . Since a is taken to be zero for terminal velocity, this results in \( F_{\text{gravity}} - F_{{{\text{air}}\,{\text{resistance}}}} = 0 \) , which means \( F_{\text{gravity}} = F_{{{\text{air}}\,{\text{resistance}}}} \) .

Instead, Sherin models these students’ solution as relying on the balancing symbolic form to associate the intuitive ideas of two influences in opposition to the symbol template □ = □. This way of leveraging the physical interpretation of the situation to affect how the mathematical equations are generated, used, and interpreted reflects physics disciplinary expertise and stands in contrast to just formally processing the mathematical syntax.

4.2 Encyclopedic Knowledge: Manifold Productive Resources are Used Dynamically

P-prims form a subset of the knowledge that individuals can bring to bear in understanding physical situations. In the resources framework, an individual’s knowledge consists of fine-grained knowledge resources. Just as some subset of our ancillary encyclopedic knowledge is applied to make meaning of language, individuals bring some subset of their resources to make meaning in physics. Different subsets of these resources can be applied to the same situation to form different meanings (just as how “the beach is safe” can be understood to either imply the safety of a beachgoer or the safety of the beach from property development). Because of the manifold possible meanings, learning physics involves refining patterns of activation of and connection to our encyclopedic knowledge base to build a coherent and stable knowledge structure that aligns with the canonical knowledge and reasoning of the discipline of physics.

Although these activations are immensely valuable for living everyday life, sometimes, when mapped into learning situations, they can be activated inappropriately or misinterpreted. One example of this is discussed in Redish ( 2014 ), (Frank 2009 , Frank and Scherr 2012 ). Students are given a set of paper tapes with dots on it. (See Fig.  5 .) The dots are made by connecting an object to the paper tape and setting the object in motion. The tape runs through a machine (Pasco Tape Timer) that taps a striking pin at a fixed rate onto the paper tape through carbon paper, making a dot every time the pin taps down as shown at the left in Fig.  5 . If the object is moving quickly, the tape moves a lot between taps and there are large spaces between the dots as shown in the sample at the top right of Fig.  5 . If the object is moving slowly, the tape moves little between taps and the dots are close together as in the bottom right of Fig.  5 .

A device for displaying the speed of motion with spaced dots and two examples

The apparatus was shown at the beginning of a recitation section, and the mechanism explained by a graduate teaching assistant. Students were working in groups, and each group of four students were given four tapes of different lengths containing six dots (as shown in Fig.  5 ). The first question in their worksheet asked, “Which tape took longer to make?” Since the pin taps at a fixed rate, any tape with six dots would take the same time to make. Multiple groups of students were seen to transition quickly from one interpretation to another. In response to the first question in the lesson, asking “Which tape took longer to maker,” many students said something like, “Obviously, it takes less time to generate the shorter tape [more closely spaced dots]… because it’ shorter.” We interpret these students as applying an easily accessible intuitive resource that “more distance implies more time” (implicitly assuming no change in velocity). A short time later, in response to a prompt asking them a more detailed question (to find the velocity), these groups reinterpret the lengths of the strips as proportional to how fast the tape was pulled through the machine, drawing on the intuitive idea that “more distance implies more speed” (implicitly assuming—correctly—no change in time).

There are two important features to note from this common reasoning trajectory. First, in different moments, a group of students can connect different cognitive resources to reach different interpretations about what the different lengths mean. Second, the groups draw on different pieces of their encyclopedic knowledge depending on how they have contexualized the task in response to the different cues in different parts of the lesson to make two different (and mutually exclusive) meanings of the same objects.

A crucial part of the resources framework is the observation that resources are general—neither right nor wrong until the context and use is specified. Although one of the students’ conclusions about the length of the ticker tapes was incorrect here, their intuitive knowledge resources cannot be defined as correct or incorrect independent of context. In linguistics, it would be wrong to interpret the sentence “Sam broke his arm, so he’s in a cast for 6 weeks” as meaning that Sam was acting in a theater performance given the context. Yet, it would be similarly incorrect to say it is wrong to think “cast” can mean “a group of actors in the same production.” With respect to the ticker tapes, although the intuition “more distance implies more time” is not correct in interpreting these ticker tape strips, it would be correct in other contexts, such as comparing airline flight times between different cities.

In the next section, we provide an example of two ways that physics students can draw on encyclopedic knowledge resources to use and interpret the equation \( v = v_{0} + at \) . We argue that although both meanings are correct ways of interpreting an equation in physics, opportunistically and productively blending physical meaning with the mathematical syntax evidences more expert-like reasoning in the discipline of physics.

4.2.1 Case Study #1: Blending Physical Meaning with Mathematical Structure

Understanding an equation in physics is not limited to connecting the symbols to physical variables and being able to perform the operations with that equation. An important component is being able to connect the mathematical operations in the equation to their physical meaning and integrating the equation with its implications in the physical world. In this section, we illustrate the differences in ways that meaning could be attached to an equation by analyzing excerpts of two clinical interviews with students in an introductory physics class for engineers (Kuo et al. 2013 ). Both excerpts focus on the students’ understanding of the equation for the velocity of an object moving with a constant acceleration: \( v = v_{0} + at \) , [Eq. ( 7 )] where v is the velocity of the object at time t , v 0 is the velocity of the object at t  = 0, and a is a constant acceleration. Both students can use the equation satisfactorily for solving problems, but the encyclopedic meaning ascribed to the equation by Alex is different than that ascribed by Pat.

In Alex’s interview, when she is asked to explain the velocity equation, she focused on the mathematical meaning of the symbols:

I: Here’s an equation; OK, you’ve probably seen it. Alex: Yeah. I: Right. So suppose you had to explain this equation to a friend from class. How would you go about doing that? Alex: Umm, okay, well, umm, I guess, first of all, well, it’s the equation for velocity. Umm, well, I would, I would tell them that it’s uh, I mean, it’s the integral of acceleration, the derivative of { furrows brow } position, right? So, that’s how they could figure it out, I don’t know. I don’t really { laughs }, I’m not too sure what else I would say about it. You can find the velocity. Like, I guess it’s interesting because you can find the velocity at any time if you have the initial velocity, the acceleration, and time…

Alex’s response suggests that she thinks about the equation as something that allows one to calculate the velocity of an object at any moment. She does refer to the velocity as the derivative of position and integral of acceleration but her comment does not reflect whether those mathematical operations connecting velocity to position and velocity to acceleration have any deeper physical meaning for her.

In Pat’s interview, she also refers to the physical meaning of the symbols in the equation, but her explanation is not limited to their names. Her explanation seems to call on her physical knowledge about motion, units, and processes of change:

I: Ok. So here’s probably an equation that you have seen before, right? And um, if you were to explain this equation to a friend from class, how would you go about explaining this? Pat: Well, I think the first thing you’d need to go over would be the definitions of each variable and what each one means, and I guess to get the intuition part, I’m not quite sure if I would start with dimensional analysis or try to explain each term before that. Because I mean if you look at it from the unit side, it’s clear that acceleration times time is a velocity, but it might be easier if you think about, you start from an initial velocity and then the acceleration for a certain period of time increases that or decreases that velocity.

Pat also looks at the physical meaning of each symbol in the equation and how they are connected. She brings in knowledge about units and how the dimensions in the equation must be consistent between terms. But she lends a deeper meaning to the equation by bringing in additional knowledge about the physical process of acceleration that changes the initial velocity. Further excerpts of her interview show that, for Pat, the mathematical formulation of Eq. ( 7 ) is stably connected to the conceptual schema of base  +  change , where you start with an initial quantity and then something changes it. The “ at” term for her reflects the total amount by which the initial value is changed. We model her reasoning as connecting the base  +  change symbolic form to relationships between the physical variables of velocity and acceleration to make meaning of this equation.

In Sect.  2 , we argue that the cultural norms of the discipline of physics include opportunistically blending in physical meaning in using and interpreting equations. Because of this, we argue here that Pat’s use of base  +  change to blend in an overall conceptual story relating velocity and acceleration onto the velocity equation reflects more expert-like physics practice.

Furthermore, we see that this interpretation of the equation affects its use in problem solving. Later in the interview, Alex and Pat use the equation to solve a problem about differences in the speeds of two balls thrown from a building at the same time with different initial velocities. Alex uses the equation as a tool to compute final velocities given the initial velocity, time, and acceleration. Pat on the other hand uses the equation much more as an expert physicist might, reaching the answer without needing to plug in numbers and carry out arithmetic calculations, and she exhibits an expert-like understanding of why the result should be what it is, based on the meaning that she assigns to the structure of the equation.

4.3 Contextualization: Activation Depends on Conceptual, Epistemological, and Affective Factors

A human mind contains a vast amount of knowledge about many things but has limited ability to access that knowledge at any given time. (Miller 1956 ) As cognitive semanticists point out, context matters significantly in how stimuli are interpreted and this is as true in a physics class as in everyday life. Sometimes, this means that structural features of the problem context cue knowledge related to physics concepts—a problem with a block on a ramp cues the conceptual knowledge of forces and motion rather than conceptual knowledge related to magnetic fields. Yet, context extends beyond the physics content of the particular problems presented. It includes the context of being a student in a physics class interacting with teachers and other students. Students bring a multitude of resources related to “ways-of-knowing” to physics classes, developed through years of experience with schooling and with knowledge building. Since “epistemology” means the science of knowing about knowledge, we refer to these as epistemological resources .

Elby and Hammer ( 2001 ) and Hammer and Elby ( 2002 , 2003 ) explore what basic ideas people use to decide they know something. Some basic epistemological resources that are relevant for our consideration of mathematics use in science include (Bing and Redish 2009 , 2012 ):

Embodied physical intuition: Knowledge constructed from experience and perception (e.g., p-prims) is reliable.

By authority: Information from an authoritative source can be trusted.

Calculation: Algorithmic computational steps lead to a trustable result.

Physical mapping to math: A mathematical symbolic representation faithfully characterizes some feature(s) of the physical or geometric systems it is intended to represent.

Mathematical consistency: Mathematical representations have a regularity and reliability that are consistent across different situations.

We can model the encyclopedic knowledge that individuals bring to bear to make contextual meaning as coordinated assemblies of resources—the coordination including conceptual, epistemological, and affective factors among others. As a result of students’ experience, both in their everyday lives and in their schooling, we note that often these coordinations of resources may develop into regular, aligned patterns. For example, if mathematical processing is frequently activated with a negative and stressful affect— this is distressingly hard —they might tend to activate together.

Although we have thus far emphasized the manifold possible patterns of resource activation, we also believe that sometimes expectations about knowledge and learning can be quite stable and reliably cued. Alignment can become strong, at least for a while and in common contexts. In asking how to transform T ( x , y ) to T ( r , θ ), we expect physics and mathematics disciplinary experts to tend to answer according to the rules of their particular disciplinary contexts. In physics, students may spend 15 minutes or longer, stuck using unproductive knowledge-building methods to solve a problem. (Bing and Redish 2008 ) Moreover, a student in a physics class may bring to bear ways of making meaning in physics—problem solving or studying strategies—that consistently fail to be useful, even though there is evidence that they possess the resources for more productive strategies (Hammer et al. 2005 ). The primacy and stability of particular patterns of aligned activations can inhibit transitions to other potentially more productive ones.

However, it is a mistake to assume that the apparent stability of one line of reasoning implies no alternatives. Just as with language, making meaning in physics is a dynamic process, often responding to shifting social situations and cues. Through the next case study, we make the point that even apparently stable modes of reasoning can be re-evaluated and shifted by an interviewer offering appropriate cues.

4.3.1 Case Study #2: How Shifts in Epistemology Support Shifts in the Meaning of Equations

As an example of the role that epistemology can play in the dynamics of meaning-making in physics, we present the case study of one student’s changing interpretations of two “isomorphic” equations (Kuo 2013 ).

Devon was interviewed the summer after his first-semester physics course. His interview started with the same prompt that Alex and Pat answered: explain the equation \( v = v_{0} + at \) . Devon’s explanation, targeted to a middle school student, was similar to Alex’s, focusing on the meaning of the variables only as much as required for using the equation as a computational tool.

Yet, we expect that Devon, an undergraduate engineering major, possesses resources for making sense of equations with base  +  change and other symbolic forms. After the velocity prompt, Devon was asked to come up with an isomorphic equation for money:

You start out with m 0 dollars, and you make r dollars per day. How many dollars ( m ) would you have at the end of d working days? Could you express the number of dollars ( m ) in an equation?

Devon quickly comes up with the correct equation: \( m = m_{0} + rd \) . When asked to explain to a middle school student how he got this equation, he explains through base  +  change, starting with the more concrete example of how much money you would make in 1 week:

Devon: So, if you work five days and you get so much money per each day, what do you do to calculate your total earnings for those five days? And I think by twelve they would know, oh just multiply by how much you get per day, OK, and that’s going to take care of this, the r times d, and then you know to get the total, they already know what you start off with so they would know to add it to that. I: Why add? Devon: Because you want to have the total, like for, you start off with a certain amount, and you want to know how much you have after the week, so your initial amount plus how much you earned that week is equal to the total amount of money that you have.

As with Pat’s explanation of \( v = v_{0} + at \) , Devon’s explanation of the money equation identifies the quantities of base (“you start off with a certain amount”), change (“how much you earned that week”), and an overall story that relates base and change to final amount (“your initial amount plus how much you earned that week is equal to the total amount of money that you have”). There are many conceptual differences in the problems that contribute to cuing Devon’s different interpretations of the two equations. Here, we highlight an epistemological difference. In the interview Devon views understanding equations in physics as memorization and understanding equations in math as understanding why those equations work:

I: When do you feel really comfortable with an equation, when do you feel that you really understand an equation? Devon: Well, in physics, I feel comfortable when I memorize the thing, and I know all the units that are attached to it. ‘Cuz as I said, I like, I’m a concrete sequential kind of guy in the math, so if I know, if I could see that the units make sense, then I know what I’m doing must be right, I don’t, I just don’t like thinking of the concepts behind it, I don’t like thinking of gravity, you know. Other people think because of this, because of gravity, it’s going to do such and such, I’m not, you know, I like just focusing on units and just if it makes sense, and I just memorize the equation, I mean, other people can derive the equations by, I don’t know, Newton’s second law or doing the free body diagrams and they can derive an equation or a certain kinematic-, but I don’t do that, I just think of what makes sense, unit wise, I guess. I: And what about math? When do you feel that you really understand an equation in math? Devon: Well, in math, well like, there’s so many proofs, and it just makes sense in my mind, I don’t know, like derivatives and integrals and Jacobian transformations. It just all makes sense to me, because there’s a reason it works, and it’s just one reason. It’s not like in physics really where there’s so many different cases like I said before. In math, like if I understand the proofs of why it’s that way, and then, I’m comfortable using that equation.

Here, two explicit, in-the-moment epistemological stances toward equations emerge. In physics, Devon prefers focusing on units and memorizing the equation rather than thinking about the concepts behind the equations. He acknowledges that other people can reason about concepts (like gravity) or derive equations from more basic principles (like Newton’s second law), but he explicitly chooses not to.

In contrast, equations in math make sense to Devon. Unlike the multiple different cases in physics, math for Devon is simpler and unified (“There’s one reason it works, and it’s just one reason.”). Perhaps because of the perceived relative simplicity, Devon seeks understanding of math equations, whereas the proliferation of equations and concepts in physics make that same kind of understanding seem too difficult to attain.

Later in the interview, Devon moves back to the physics domain to explain an equation for the speed of a car with base  +  change . This suggests that explaining the money equation not only revealed how Devon makes sense of equations in math problem contexts, but also tapped into a productive resource that he could then, with little effort, apply to make meaning with an equation in a physics context.

Devon’s reasoning provides examples of two key features of the resources framework. First, his original failure to use base  +  change does not represent a knowledge deficit. Devon’s reasoning with equations is context dependent and another problem context reveals how he can draw on base  +  change to interpret a similar equation. Second, Devon’s epistemological stances toward equations in math and physics align with and support different modes of reasoning with equations.

4.4 Contextualization: How Affect Shifts Along with Shifts in Conceptual Meaning and Epistemology

Thus far, we have discussed how the dynamics of student reasoning in physics depends on knowledge, both conceptual and epistemological. Yet, as teachers, we know that students’ affective states are as important as the knowledge they possess in determining how they make sense of physics problems. For example, as teachers, we know intuitively not to teach new material on the day of an exam because the negative affect that results will impede deep engagement with new topics.

Yet, despite this instructional intuition as well as research showing the correlations between emotions and learning, studies of physics students’ conceptual knowledge and epistemologies continue to greatly outnumber the studies that investigate the role of affect and emotions in the dynamics of those students’ reasoning. We should not take the detailed specification of conceptual and epistemological resources in the resources framework to imply that other factors, such as affect, play no role in contextualization and meaning-making. Next, we discuss a case study illustrating how affect is coupled with conceptual and epistemological resources in shifts in student reasoning.

4.4.1 Case Study #3: Student Response to Math is Dynamic and May be Mediated by Affect

Gupta and Elby ( 2011 ) report on an extremely interesting interview with a student in an introductory physics class for engineers that illustrates the dynamic response of a student trying to make sense of mathematics in a physics class and displays the interaction of conceptual, epistemological, and affective factors.

The interview occurred during the first semester of physics for engineers. The interviewer was not involved in teaching or evaluating the class. After a discussion of Eq. ( 7 ), a topic that had been discussed in class, the interviewer turned to a topic that had not yet been discussed: pressure in fluids. He presents the equation:

where p represents the pressure under the surface of a lake, p 0 is the pressure at the top of the lake, ρ is the density of water, g is the gravitational field, and h is the depth below the water’s surface. He then asks the subject, Jim, whether he has seen the equation before, and Jim replies that he has not. The interviewer then asks whether pressure 7 m beneath the surface of the lake is greater, equal to, or less than the pressure 5 m below the surface.

Jim’s first inclination is to use the equation and, after some to-ing and fro-ing, he decides that h should be negative under the surface and therefore that the pressure at 7 m below the surface should be less than that at 5 m. He is somewhat uncomfortable with this as it contradicts his intuition and experience. He has made the mistake of taking h as negative (taking positive as upward) but not also taking g as negative (gravity points down).

When asked how a friend from English class might reason about the problem (I: “Suppose, there was a friend of yours in English, [who] does not really know physics and questions, kind of thing. Could they have answered this question?”), Jim demonstrates he possesses a good physical intuition about the situation.

Jim: Like, if they have actually been under water, so the pressure, they might know a little bit about pressure under water…Like a rough estimate. The pressure was higher when I was deeper. [I: Okay] The pressure was lower when I was higher to the surface … They could argue from their personal experience like, one time I was scuba diving and I was like 30 feet below the water and pressure was like, pressure was very high. Like I was just swimming, I was just couple of feet below the water and the pressure was not that much.

But he sticks firmly to his trust of the equation. He explains he is in a physics class, states that “this is hard,” and expresses the stance that on an exam he would trust his analysis of the equation over his intuition.

Jim: For an equation to be given to you, it has to be like theory and it has to be fact-bearing. So, fact applies for everything. It is like a law. It applies to every single situation you could be in. But, like, your experience at times or perception is just different—or you don’t have the knowledge of that course or anything. So, I will go with the people who have done the law and it has worked time after time after time.

Although Jim can use his embodied experience effectively and correctly, his affect—his lack of confidence about his intuitions and sense that “physics is hard”—leads him to suppress an epistemological resource supporting “knowledge constructed from perception” and to rely instead on knowledge from authority and knowledge constructed from mathematical manipulation. This is shown in Fig.  6 a with the lines with arrows indicating activation and the lines with circles on the end indicating inhibition. He appears to have actively rejected his intuition. (Although the details of this figure are different from Gupta and Elby ( 2011 ), we align with them to the degree that our diagram aims to model how two different epistemological stances can share particular epistemological resources in common and to show a mechanism for how a contextual cue can shift one local epistemological stance to another.)

I: Do you think the [hydrostatic pressure] equation relates to [the physical experience of pressure]? Jim: Probably somehow, but not directly. I think there is some way that just completely links the two together, but itis not obvious what that relation is.

A bit later in the interview, the interviewer hints that perhaps (if he is going to insist that down is the negative direction) that he might consider the sign of g .

I: What do you think about g in [the hydrostatic pressure] equation? Should that be minus ten or plus ten? Jim: Oh! minus ten … So, that gives you a positive thing. [I: Okay.] I would say that the negative does not matter anymore. Oooh! I see. The higher you go under water, uh, the lower you go under water the more your pressure is, because the negative and the negative cancel out … So, the more under water you are the higher your pressure is going to be, I think now. I forgot to factor in g . That’s what I think. I: Okay. Is that more comfortable or less comfortable? Jim: That is more comfortable because it actually makes more sense to me now. And now your experience actually does work because from your experience being under water you felt more pressure as opposed to the surface. If I take into consideration both negatives, it is just positive, they just add up.

Jim’s epistemological stance has shifted dramatically, along with his affect. Now, his calculation (tied to the epistemological resources “knowledge constructed from mathematical manipulation” and “knowledge from authority”) is consistent with his intuition (“knowledge constructed from perception”) as shown in Fig.  6 b.

The interaction of Jim’s affect and epistemological resources with the physics context. a Before reconciliation; b after reconciliation. Faded nodes are temporarily inactive. (After Gupta and Elby 2011 )

Overall, the case studies that we have presented in this paper demonstrate that looking at meaning-making with mathematics through the lenses of embodied cognition, encyclopedic resources, and contextualization as refined by epistemological analysis gives us insight into ways in which students might not tap into their funds of productive knowledge, how the context can be shifted to reveal that productive knowledge, and how affect plays an interactive role in meaning-making dynamics.

5 Implications for Instructors

How do we interpret student difficulties with math in physics class? Common diagnoses are that students have not been taught the relevant mathematical tools in their math courses, or they have not learned those mathematical tools well enough to transfer them to their physics courses. The common solutions in either case are for physics instructors to re-teach the needed mathematical procedures and methods in the physics courses or to demand that the students take additional (or more rigorous) math courses.

Our analysis of the roles of physics and math from the point of view of language and meaning-making suggests an alternative diagnosis: Even if students have learned the relevant mathematical tools in their math courses, they still need to learn a component of physics expertise not present in math class—tying those formal mathematical tools to physical meaning.

To succeed in physics, students need not just to be fluent with mathematical processing in the context of physics, but also with the mathematical modeling of physical systems, blending physical meaning with mathematical structures, and interpreting and evaluating results. We as physics instructors must explicitly foster these components of expert physics practice to help students succeed in using math in physics.

At a finer-grained level, we are also teaching them how to integrate their everyday embodied and conceptual knowledge with math and how to reliably activate appropriate epistemological resources. Some of these ideas have been discussed in the literature (Tuminaro and Redish 2007 ; Bing and Redish 2009 , 2012 ; Kuo et al. 2013 ).

The take-away message is:

How mathematical formalism is used in the discipline of mathematics is fundamentally different from how mathematics is used in the discipline of physics – and this difference is often not obvious to students. For many of our students, it is important to explicitly help them learn to blend physical meaning with mathematical formalism.

The question then is, “How do we help them learn to blend physical meaning with mathematical formalism?” The three case studies we have presented suggest that even when we do not see it, the seeds of this blending are already there. Although we commonly see students failing to use math in physics, case study #1 suggests that some students, like Pat, are already engaging in expert-like use of equations in physics. Case studies #2 and #3 show examples where students who initially reason from the formal procedures of mathematics shift to layering physical meaning onto the math. Importantly, these shifts occurred without direct instruction, suggesting that students possess productive resources for interpreting math physically, which as instructors we need to help them identify and activate reliably.

To help students integrate physical meaning with the mathematics they are learning and using, we have to use our knowledge of what productive knowledge they possess, what can cue different knowledge elements, what epistemological resources students tend to use, how they feel about physics and math, and how all these elements interact.

While this is easy to say, it can be hard to do. Many physicists have spent years building their fluency and competency with mathematical manipulations until the blending of math with physical meaning becomes natural and often automatic. For many physicists, the easiest way to get through to meaning is to begin with the mathematical relations that come easily to them.

Because of this fluency with blending physical meaning with mathematical manipulations, we have observed physics instructors (including ourselves), teaching at both the introductory and advanced levels, stressing mathematical manipulation as the “go-to” resource for building knowledge. Sample problems done on the board and homework tasks often stress the processing step of the mathematical modeling model described in Fig.  4 , omitting the steps of modeling, evaluating, and interpreting, as well as the critical meaning-making steps of contextualizing physically and tying to physical experience and encyclopedic knowledge. While for experts a math-based approach may be aligned with “physical mapping to math,” for novices who do not yet fluently blend math and physical meaning, the emphasis on mathematical processing can support epistemologies that give primacy to formal laws, equations, and computational reliability to the exclusion of physical meaning.

We argue that instruction not only needs to highlight aspects of how expert physicists layer physical meaning onto mathematics, but also to tap into productive student resources for doing so. Here are two examples.

5.1 Can You Do It Without the Equations?

In 1992, the senior author of this paper (EFR) decided to switch his research effort to physics education (after 25 years as a theoretical physicist). To learn the ropes, he decided to spend a sabbatical year at the University of Washington, where Lillian McDermott (McDermott 1984 ) and her Physics Education Group were building the groundbreaking lessons, Tutorials in Introductory Physics (McDermott and Shaffer 1992 , 2001 ; Shaffer and McDermott 1992 ). As part of his experience in learning how to build lessons that worked, the senior author was being trained as a facilitator for the tutorial lessons. Groups of graduate students, postdocs, and faculty worked through the lesson, discussing how students might respond.

The first week’s lesson concerned electric currents and involved the now famous “batteries and bulbs” tasks. The first serious question showed the diagram in Fig.  7 and asked, “If all the bulbs in the circuit below are identical, rank the brightness of the bulbs.”

A batteries and bulbs problem (McDermott and Shaffer 1992 ). If the four bulbs are identical, rank the brightness of the bulbs

Your experienced theoretical physicist saw no problem with this. “This is a simple Kirchhoff’s law problem with two current loops. If I define a current I 1 in the left loop and I 2 in the right loop, the drops around each loop have to add to the rises, so I get two equations in two unknowns:

which are easily solved for the currents. The brightness of each bulb is just the power dissipated, I 2 R , so the ranking of the bulbs is like the ranking of the currents. Since my solution gave me I 1  = 2 I 2 , we must have A = D > B = C.”

Our theoretical physicist’s partner in the task was Richard Steinberg, then a first-year postdoc fresh out of an Applied Physics Ph.D. at Yale. He smiled and asked, “Can you do it without the equations?” Our theoretical physicist frowned, and responded, “Why should I?” Steinberg’s reply, “Perhaps your students aren’t as fluent with equations as you are” brought a reluctant attempt. The theorist found it surprisingly difficult at first. But after a while, the fog lifted and he realized it was much easier without the equations—and that the equations supported an interpretation in physical terms. Here is a physical analysis.

Current is conserved so it has to be continuous and divide at junctions (sum of currents in equals sum of currents out). So, whatever current goes through the battery has to also go through bulb A. At the junction to the second loop, the current will split. Since both paths have equal resistance, the current will split equally, so the current in B will equal the current in C. At the bottom, they come back together and yield the original current. So, the current in D is the same as in A—and twice that in B and C. This is the same result as given by the two equations in two unknowns, but it ties directly to the embodied concept of conserved physical flow.

Although experienced physicists will see exactly how this physical analysis connects to the mathematics of the Kirchoff’s law solution, students may not. Recently, one of us (EFR) was serving in the help center when a student asked for help with a circuit problem of the same level of complexity as the one shown above. He had struggled for an hour trying to set up and solve the Kirchhoff’s law loop equations as he had been taught in class. When he was introduced to the conceptual approach to using the basic principles, he lit up and was able to solve the problem quickly and easily, saying, “Why weren’t we shown this way to do it?” He would still need to bring his conceptual understanding into line with the mathematical reasoning needed to set up more complex problems, but the conceptual base made sense to him as a starting point in a way that the algorithmic math did not.

Instructionally, for many students, the first step needs to be highlighting the physical meaning, as Steinberg did by asking, “Can you do it without the equations?” However, the instruction is incomplete until these physical interpretations are tied back to the formal mathematical laws. For example, the conceptual idea that currents are continuous and that the voltage drop across each resistor should sum to the voltage supplied by the battery is represented mathematically through the procedure of writing the voltage changes around a closed loop and by interpreting the final equation with the parts - of - a - whole symbolic form.

The benefits of this physical meaning here are twofold: (1) The procedure of Kirchoff’s law analysis is no longer a brittle, rote procedure, but it has meaning. It is backed up by links to encyclopedic knowledge about the physical system, and (2) it may yield conceptual shortcuts in complicated circuits where setting up the equations for multiple loops may be more challenging.

5.2 Tapping into Physical Intuition for Understanding Mathematics

The senior author has also had multiple opportunities to observe the instruction of a number of physics faculty. A particularly interesting example occurred in a class of mostly biology majors in which students were learning to read potential energy graphs and relate them to forces. The students were shown Fig.  8 a and asked the question: “If two atoms have a total energy E shown by the horizontal line labeled “E”, and the potential energy interaction is as shown by the curve in the figure, when they are at a separation indicated by C, is the force between them attractive or repulsive?”

The students were still at an early stage in learning to read potential energy diagrams and struggled with the question. In two different classes, two different instructors essentially responded with the argument, “Remember the equation shown on the board [Eq. ( 10 )]. Since the derivative of the PE curve at C is positive [the slope of the tangent line in Fig.  8 (b)], the force is negative, which means it’s attractive.” [Both instructors wrote Eq. ( 10 ) on the board, but neither actually drew Fig.  8 b.]

Again, for students who have not developed facility in reading physical meaning from mathematical expressions, they may make sense of this lesson with the epistemological resources, by trusted authority (the remembered equation) and calculation .

A more effective approach for this population might be to begin with an embodied analogy and implicitly supporting epistemologies valuing physical intuition. Start with treating a potential energy curve as a track or hill and, using the analogy of gravitational PE, then place a ball on the hill as shown in Fig.  8 c. Which way will it roll? In conversations with a number of these students, none had any difficulty in coming up with the correct answer. Then, inferring that the force was to the left, they were able to see that the force to the left implied a positive slope for the PE via Eq. ( 10 ) and were then easily able to explain how that could be seen from the graph and the equation.

Three figures illustrating different epistemological approaches to an explanation. a The figure shown in the problem; b an explanation based on a formula; c an explanation based on a physical analogy

Two plausible models of epistemological resources activated for the examples in 5.1 and 5.2. The one on the left models what resources physicists may activate when starting from formal equations. The one on the right models a chain that may be more useful in helping introductory students learn the value of math in science

5.3 Teaching Physics Standing on Your Head

In the two examples discussed in Sects.  5.1 and 5.2 , one could model the physicist’s active epistemologies in the two competing instructional sequences as connected chains of epistemological resources, as displayed in Fig.  9 . The one at the left starts from valuing formalized equations and principles. The one at the right gives primacy to physical intuition and embodied experience. In many instructional situations (especially with novice or non-physics major students), the sequence on the right can be more effective for supporting the blending of physical meaning and mathematics, despite the comfort that an experienced physicist might have with the one on the left. There are two reasons for this.

The first is that we want to provide students with experience with the physics disciplinary practice of physically making meaning with mathematics. Starting from the mathematical laws of physics, while efficient, may not make the underlying physical meaning in those expressions visible to novices. From the two examples above, starting from the physical meaning and then explicitly mapping this meaning to the mathematics can help make this connection explicit for physics students (as well as instructors) for particular topics and help students see how to make this connection more generally.

The second is that, in physics, we want students to learn to take an epistemological stance that coherence between physical meaning and mathematical formalism is valuable and productive. Although in particular moments, experts may appeal to formalized physics laws and equations, we believe that that those same experts, in other moments, regularly blend physical meaning with the mathematics. However, novice students, even those capable in math, have to learn to the language of associating physical meaning with math. Because the connection between the physical meaning and the mathematics is more opaque (especially for novices) in the left epistemological chain, students may see it as an example where physics class values knowledge from authority over physical intuition. This has the instructional consequence of encouraging a novice epistemological stance (memorizing equations without understanding) even while facilitating and quickly generating a “correct” answer.

In both of our examples in this section, experienced physicists’ epistemologies could be plausibly modeled through the sequence on the left, and, for them, we expect that this sequence is stable and reliable. This is likely because this sequence can be very productive in a physics researcher’s career. Yet, the stability of this instructional sequence may also work to inhibit the activation of other sequences in instruction that may better support the student epistemologies we believe are productive for making sense of math in physics.

It might well be preferable to “teach physics standing on your head” by beginning with the physical meaning and creating a chain of association to the math, both strengthening the students’ skills of “seeing physical meaning” in equations and helping them develop the epistemological stance that equations in physics should be interpreted physically.

We suggest that an analysis of instructional success and activity building using this sort of analysis might prove productive in research and in the building of instructional materials. (See, for example, Elby 2001 ; Gupta and Elby 2011 ; Bing and Redish 2012 ; Kuo et al. 2013 ).

6 Conclusion

In a standard interpretation, lack of success with math in physics is attributed to a failure to transfer mathematical skills from math class to physics class. The considerations raised here suggest an alternative possible diagnosis: Learning math in math class and math in physics class should be treated as learning two related but distinct languages: Although there is significant overlap, there are also important differences, and expertise in one does not guarantee expertise in another. Although both physics and math make meaning in the way that any language does—by building on physical experience, by drawing on broad (encyclopedic) knowledge, and by contextualizing—they do so in different ways. We must recognize that we are asking students to become bilingual and provide instruction and practice in blending in physical interpretation on the mathematical formalism. This perspective raises significant issues both for research and for instruction.

Parts of this paper are based on contributions to conference proceedings (Redish 2005 ; Redish and Gupta 2010 ).

A “shibboleth” is a word or phrase that can be used to distinguish members of a group from non-members.

Mathematicians use an abstract analog of a physical function in classes such as manifold theory, but it is only commonly used in math classes at an advanced level.

One of us (EFR) was told in a college math class for math majors that using the same symbol to represent different functional dependences was “an abuse of notation.”

Of course, only parts of these experiences are extracted and abstracted in building abstract concepts. To see more detail of how these “partial analogies” are used to build conceptual meaning into rigorous concepts in mathematics, see Fauconnier and Turner ( 2003 ) and Alexander ( 2012 ).

It is interesting to note that these ideas are foreshadowed in the highly influential work of Poincare ( 1905 ).

Although this is the ideal, many mathematicians argue that there are subtle issues that prevent this ideal from ever being realized, even in principle. (Goldstein 2006 ; Carroll 1897 ).

We note that the order of what is interpreted as the “base” and “change” is reversed in the way our two examples are usually presented. We are not aware of any research on the implications of such an exchange for an interpretation of the symbolic form, but for our purposes base   +   change and change   +   base are conceptually identical interpretations, if not identical in grammatical structure.

Evans and Green ( 2006 ) refer to this term as conceptualization . We have chosen a different term for two reasons: to focus on the fact that this process is the response to context and to differentiate from the different use of the word concept by physics education researchers.

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Acknowledgments

We are grateful for discussions with Andrew Elby and Ayush Gupta on the topic of this paper. We would like to thank the members of the University of Maryland Physics and Biology Education Research Groups for many relevant conversations. This work was supported in part by a grant from the US National Science foundation.

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• 14 May 2024

Why mathematics is set to be revolutionized by AI

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Giving birth to a conjecture — a proposition that is suspected to be true, but needs definitive proof — can feel to a mathematician like a moment of divine inspiration. Mathematical conjectures are not merely educated guesses. Formulating them requires a combination of genius, intuition and experience. Even a mathematician can struggle to explain their own discovery process. Yet, counter-intuitively, I think that this is the realm in which machine intelligence will initially be most transformative.

In 2017, researchers at the London Institute for Mathematical Sciences, of which I am director, began applying machine learning to mathematical data as a hobby. During the COVID-19 pandemic, they discovered that simple artificial intelligence (AI) classifiers can predict an elliptic curve’s rank 1 — a measure of its complexity. Elliptic curves are fundamental to number theory, and understanding their underlying statistics is a crucial step towards solving one of the seven Millennium Problems, which are selected by the Clay Mathematics Institute in Providence, Rhode Island, and carry a prize of US$1 million each. Few expected AI to make a dent in this high-stakes arena.

AI now beats humans at basic tasks — new benchmarks are needed, says major report

AI has made inroads in other areas, too. A few years ago, a computer program called the Ramanujan Machine produced new formulae for fundamental constants 2 , such as π and e . It did so by exhaustively searching through families of continued fractions — a fraction whose denominator is a number plus a fraction whose denominator is also a number plus a fraction and so on. Some of these conjectures have since been proved, whereas others remain open problems.

Another example pertains to knot theory, a branch of topology in which a hypothetical piece of string is tangled up before the ends are glued together. Researchers at Google DeepMind, based in London, trained a neural network on data for many different knots and discovered an unexpected relationship between their algebraic and geometric structures 3 .

How has AI made a difference in areas of mathematics in which human creativity was thought to be essential?

First, there are no coincidences in maths. In real-world experiments, false negatives and false positives abound. But in maths, a single counterexample leaves a conjecture dead in the water. For example, the Pólya conjecture states that most integers below any given integer have an odd number of prime factors. But in 1960, it was found that the conjecture does not hold for the number 906,180,359. In one fell swoop, the conjecture was falsified.

Second, mathematical data — on which AI can be trained — are cheap. Primes, knots and many other types of mathematical object are abundant. The On-Line Encyclopedia of Integer Sequences (OEIS) contains almost 375,000 sequences — from the familiar Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, ...) to the formidable Busy Beaver sequence (0, 1, 4, 6, 13, …), which grows faster than any computable function. Scientists are already using machine-learning tools to search the OEIS database to find unanticipated relationships.

DeepMind AI outdoes human mathematicians on unsolved problem

AI can help us to spot patterns and form conjectures. But not all conjectures are created equal. They also need to advance our understanding of mathematics. In his 1940 essay A Mathematician’s Apology , G. H. Hardy explains that a good theorem “should be one which is a constituent in many mathematical constructs, which is used in the proof of theorems of many different kinds”. In other words, the best theorems increase the likelihood of discovering new theorems. Conjectures that help us to reach new mathematical frontiers are better than those that yield fewer insights. But distinguishing between them requires an intuition for how the field itself will evolve. This grasp of the broader context will remain out of AI’s reach for a long time — so the technology will struggle to spot important conjectures.

But despite the caveats, there are many upsides to wider adoption of AI tools in the maths community. AI can provide a decisive edge and open up new avenues for research.

Mainstream mathematics journals should also publish more conjectures. Some of the most significant problems in maths — such as Fermat’s Last Theorem, the Riemann hypothesis, Hilbert’s 23 problems and Ramanujan’s many identities — and countless less-famous conjectures have shaped the course of the field. Conjectures speed up research by pointing us in the right direction. Journal articles about conjectures, backed up by data or heuristic arguments, will accelerate discovery.

Last year, researchers at Google DeepMind predicted 2.2 million new crystal structures 4 . But it remains to be seen how many of these potential new materials are stable, can be synthesized and have practical applications. For now, this is largely a task for human researchers, who have a grasp of the broad context of materials science.

Similarly, the imagination and intuition of mathematicians will be required to make sense of the output of AI tools. Thus, AI will act only as a catalyst of human ingenuity, rather than a substitute for it.

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For everyone whose relationship with mathematics is distant or broken, Jo Boaler , a professor at Stanford Graduate School of Education (GSE), has ideas for repairing it. She particularly wants young people to feel comfortable with numbers from the start – to approach the subject with playfulness and curiosity, not anxiety or dread.

“Most people have only ever experienced what I call narrow mathematics – a set of procedures they need to follow, at speed,” Boaler says. “Mathematics should be flexible, conceptual, a place where we play with ideas and make connections. If we open it up and invite more creativity, more diverse thinking, we can completely transform the experience.”

Boaler, the Nomellini and Olivier Professor of Education at the GSE, is the co-founder and faculty director of Youcubed , a Stanford research center that provides resources for math learning that has reached more than 230 million students in over 140 countries. In 2013 Boaler, a former high school math teacher, produced How to Learn Math , the first massive open online course (MOOC) on mathematics education. She leads workshops and leadership summits for teachers and administrators, and her online courses have been taken by over a million users.

In her new book, Math-ish: Finding Creativity, Diversity, and Meaning in Mathematics , Boaler argues for a broad, inclusive approach to math education, offering strategies and activities for learners at any age. We spoke with her about why creativity is an important part of mathematics, the impact of representing numbers visually and physically, and how what she calls “ishing” a math problem can help students make better sense of the answer.

What do you mean by “math-ish” thinking?

It’s a way of thinking about numbers in the real world, which are usually imprecise estimates. If someone asks how old you are, how warm it is outside, how long it takes to drive to the airport – these are generally answered with what I call “ish” numbers, and that’s very different from the way we use and learn numbers in school.

In the book I share an example of a multiple-choice question from a nationwide exam where students are asked to estimate the sum of two fractions: 12/13 + 7/8. They’re given four choices for the closest answer: 1, 2, 19, or 21. Each of the fractions in the question is very close to 1, so the answer would be 2 – but the most common answer 13-year-olds gave was 19. The second most common was 21.

I’m not surprised, because when students learn fractions, they often don’t learn to think conceptually or to consider the relationship between the numerator or denominator. They learn rules about creating common denominators and adding or subtracting the numerators, without making sense of the fraction as a whole. But stepping back and judging whether a calculation is reasonable might be the most valuable mathematical skill a person can develop.

But don’t you also risk sending the message that mathematical precision isn’t important?

I’m not saying precision isn’t important. What I’m suggesting is that we ask students to estimate before they calculate, so when they come up with a precise answer, they’ll have a real sense for whether it makes sense. This also helps students learn how to move between big-picture and focused thinking, which are two different but equally important modes of reasoning.

Some people ask me, “Isn’t ‘ishing’ just estimating?” It is, but when we ask students to estimate, they often groan, thinking it’s yet another mathematical method. But when we ask them to “ish” a number, they're more willing to offer their thinking.

Ishing helps students develop a sense for numbers and shapes. It can help soften the sharp edges in mathematics, making it easier for kids to jump in and engage. It can buffer students against the dangers of perfectionism, which we know can be a damaging mindset. I think we all need a little more ish in our lives.

You also argue that mathematics should be taught in more visual ways. What do you mean by that?

For most people, mathematics is an almost entirely symbolic, numerical experience. Any visuals are usually sterile images in a textbook, showing bisecting angles, or circles divided into slices. But the way we function in life is by developing models of things in our minds. Take a stapler: Knowing what it looks like, what it feels and sounds like, how to interact with it, how it changes things – all of that contributes to our understanding of how it works.

There’s an activity we do with middle-school students where we show them an image of a 4 x 4 x 4 cm cube made up of smaller 1 cm cubes, like a Rubik’s Cube. The larger cube is dipped into a can of blue paint, and we ask the students, if they could take apart the little cubes, how many sides would be painted blue? Sometimes we give the students sugar cubes and have them physically build a larger 4 x 4 x 4 cube. This is an activity that leads into algebraic thinking.

Some years back we were interviewing students a year after they’d done that activity in our summer camp and asked what had stayed with them. One student said, “I’m in geometry class now, and I still remember that sugar cube, what it looked like and felt like.” His class had been asked to estimate the volume of their shoes, and he said he’d imagined his shoes filled with 1 cm sugar cubes in order to solve that question. He had built a mental model of a cube.

When we learn about cubes, most of us don’t get to see and manipulate them. When we learn about square roots, we don’t take squares and look at their diagonals. We just manipulate numbers.

I wonder if people consider the physical representations more appropriate for younger kids.

That’s the thing – elementary school teachers are amazing at giving kids those experiences, but it dies out in middle school, and by high school it’s all symbolic. There’s a myth that there’s a hierarchy of sophistication where you start out with visual and physical representations and then build up to the symbolic. But so much of high-level mathematical work now is visual. Here in Silicon Valley, if you look at Tesla engineers, they're drawing, they're sketching, they're building models, and nobody says that's elementary mathematics.

There’s an example in the book where you’ve asked students how they would calculate 38 x 5 in their heads, and they come up with several different ways of arriving at the same answer. The creativity is fascinating, but wouldn’t it be easier to teach students one standard method?

A depiction of various ways to calculate 38 x 5, numerically and visually. | Courtesy Jo Boaler

That narrow, rigid version of mathematics where there’s only one right approach is what most students experience, and it’s a big part of why people have such math trauma. It keeps them from realizing the full range and power of mathematics. When you only have students blindly memorizing math facts, they’re not developing number sense. They don’t learn how to use numbers flexibly in different situations. It also makes students who think differently believe there’s something wrong with them.

When we open mathematics to acknowledge the different ways a concept or problem can be viewed, we also open the subject to many more students. Mathematical diversity, to me, is a concept that includes both the value of diversity in people and the diverse ways we can see and learn mathematics. When we bring those forms of diversity together, it’s powerful. If we want to value different ways of thinking and problem-solving in the world, we need to embrace mathematical diversity.

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May 21, 2024

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The case for 'math-ish' thinking

by Stanford University

For everyone whose relationship with mathematics is distant or broken, Jo Boaler, a professor at Stanford Graduate School of Education (GSE), has ideas for repairing it. She particularly wants young people to feel comfortable with numbers from the start—to approach the subject with playfulness and curiosity, not anxiety or dread.

"Most people have only ever experienced what I call narrow mathematics—a set of procedures they need to follow, at speed," Boaler says. "Mathematics should be flexible, conceptual, a place where we play with ideas and make connections. If we open it up and invite more creativity, more diverse thinking, we can completely transform the experience."

Boaler, the Nomellini and Olivier Professor of Education at the GSE, is the co-founder and faculty director of Youcubed, a Stanford research center that provides resources for math learning that has reached more than 230 million students in over 140 countries. In 2013 Boaler, a former high school math teacher, produced How to Learn Math, the first massive open online course (MOOC) on mathematics education . She leads workshops and leadership summits for teachers and administrators, and her online courses have been taken by over a million users.

In her new book, " Math-ish: Finding Creativity, Diversity, and Meaning in Mathematics ," Boaler argues for a broad, inclusive approach to math education, offering strategies and activities for learners at any age. We spoke with her about why creativity is an important part of mathematics, the impact of representing numbers visually and physically, and how what she calls "ishing" a math problem can help students make better sense of the answer.

What do you mean by 'math-ish' thinking?

It's a way of thinking about numbers in the real world, which are usually imprecise estimates. If someone asks how old you are, how warm it is outside, how long it takes to drive to the airport—these are generally answered with what I call "ish" numbers, and that's very different from the way we use and learn numbers in school.

In the book I share an example of a multiple-choice question from a nationwide exam where students are asked to estimate the sum of two fractions: 12/13 + 7/8. They're given four choices for the closest answer: 1, 2, 19, or 21. Each of the fractions in the question is very close to 1, so the answer would be 2—but the most common answer 13-year-olds gave was 19. The second most common was 21.

I'm not surprised, because when students learn fractions, they often don't learn to think conceptually or to consider the relationship between the numerator or denominator. They learn rules about creating common denominators and adding or subtracting the numerators, without making sense of the fraction as a whole. But stepping back and judging whether a calculation is reasonable might be the most valuable mathematical skill a person can develop.

But don't you also risk sending the message that mathematical precision isn't important?

I'm not saying precision isn't important. What I'm suggesting is that we ask students to estimate before they calculate, so when they come up with a precise answer, they'll have a real sense for whether it makes sense. This also helps students learn how to move between big-picture and focused thinking, which are two different but equally important modes of reasoning.

Some people ask me, "Isn't 'ishing' just estimating?" It is, but when we ask students to estimate, they often groan, thinking it's yet another mathematical method. But when we ask them to "ish" a number, they're more willing to offer their thinking.

Ishing helps students develop a sense for numbers and shapes. It can help soften the sharp edges in mathematics, making it easier for kids to jump in and engage. It can buffer students against the dangers of perfectionism, which we know can be a damaging mindset. I think we all need a little more ish in our lives.

You also argue that mathematics should be taught in more visual ways. What do you mean by that?

For most people, mathematics is an almost entirely symbolic, numerical experience. Any visuals are usually sterile images in a textbook, showing bisecting angles, or circles divided into slices. But the way we function in life is by developing models of things in our minds. Take a stapler: Knowing what it looks like, what it feels and sounds like, how to interact with it, how it changes things—all of that contributes to our understanding of how it works.

There's an activity we do with middle-school students where we show them an image of a 4 x 4 x 4 cm cube made up of smaller 1 cm cubes, like a Rubik's Cube. The larger cube is dipped into a can of blue paint, and we ask the students, if they could take apart the little cubes, how many sides would be painted blue? Sometimes we give the students sugar cubes and have them physically build a larger 4 x 4 x 4 cube. This is an activity that leads into algebraic thinking.

Some years back we were interviewing students a year after they'd done that activity in our summer camp and asked what had stayed with them. One student said, "I'm in geometry class now, and I still remember that sugar cube, what it looked like and felt like." His class had been asked to estimate the volume of their shoes, and he said he'd imagined his shoes filled with 1 cm sugar cubes in order to solve that question. He had built a mental model of a cube.

When we learn about cubes, most of us don't get to see and manipulate them. When we learn about square roots, we don't take squares and look at their diagonals. We just manipulate numbers.

I wonder if people consider the physical representations more appropriate for younger kids.

That's the thing—elementary school teachers are amazing at giving kids those experiences, but it dies out in middle school, and by high school it's all symbolic. There's a myth that there's a hierarchy of sophistication where you start out with visual and physical representations and then build up to the symbolic. But so much of high-level mathematical work now is visual. Here in Silicon Valley, if you look at Tesla engineers, they're drawing, they're sketching, they're building models, and nobody says that's elementary mathematics.

There's an example in the book where you've asked students how they would calculate 38 x 5 in their heads, and they come up with several different ways of arriving at the same answer. The creativity is fascinating, but wouldn't it be easier to teach students one standard method?

That narrow, rigid version of mathematics where there's only one right approach is what most students experience, and it's a big part of why people have such math trauma. It keeps them from realizing the full range and power of mathematics. When you only have students blindly memorizing math facts, they're not developing number sense.

They don't learn how to use numbers flexibly in different situations. It also makes students who think differently believe there's something wrong with them.

When we open mathematics to acknowledge the different ways a concept or problem can be viewed, we also open the subject to many more students. Mathematical diversity, to me, is a concept that includes both the value of diversity in people and the diverse ways we can see and learn mathematics.

When we bring those forms of diversity together, it's powerful. If we want to value different ways of thinking and problem-solving in the world, we need to embrace mathematical diversity.

Provided by Stanford University

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