Read an Excerpt
CHAPTER 1
History in the Mathematics Classroom
Where did mathematics come from? Has arithmetic always worked the way you learned it in school? Could it work any other way? Who thought up all those rules of algebra, and why did they do it? What about the facts and proofs of geometry?
Mathematics is an ongoing human endeavor, like literature, physics, art, economics, or music. It has a past and a future, as well as a present. The mathematics we learn and use today is in many ways very different from the mathematics of 1000, or 500, or even 100 years ago. In the 21st century it will no doubt evolve further. Learning about math is like getting to know another person. The more you know of someone's past, the better able you are to understand and interact with him or her now and in the future.
To learn mathematics well at any level, you need to understand the relevant questions before you can expect the answers to make sense. Understanding a question often depends on knowing the history of an idea. Where did it come from? Why is or was it important? Who wanted the answer and what did t hey want it for? Each stage in the development of mathematics builds on what has come before. Each contributor to that development was (or is) a person with a past and a point of view. How and why they thought about what they did is often a critical ingredient in understanding their contribution.
To teach mathematics well at any level, you need to help your students see the underlying questions and thought patterns that knit the details together. This attention to such questions and patterns is a hallmark of the best curricula for school mathematics. It is the driving force behind the Standards for Mathematical Practice, a major component in all levels of the Common Core State Standards. It is also reflected in the National Research Council's "Framework for K-12 Science Education" section of their 2013 report, The Mathematical Sciences in 2025. Most students, especially in the early grades, are naturally curious about where things come from. With your help, that curiosity can lead them to make sense of the mathematical processes they need to know.
So what's a good way to use history in the math classroom? The first answer that comes to mind is "storytelling" — historical anecdotes, or, more generally, biographical information. Here's a typical scene. When introducing the idea of how to sum an arithmetic progression, the teacher tells a story about Carl Fried rich Gauss.
When he was about 10 years old (some versions of the story say 7), Gauss's teacher gave the class a long assignment, apparently to carve out some peace and quiet for him self. The assignment was to add all the numbers from 1 to 100. The class star ted working away on t heir slates, but young Gauss simply wrote 5050 on his slate and said "There it is." The astonished teacher assumed Gauss h ad simply guessed, and, not knowing the right answer him self, told Gauss to keep quiet until the others were done, and then they would see who was right. To his surprise, the answer that the others got was also 5050, showing that young Gauss was correct. How had he done it?
Telling such a story achieves some useful things. It is, after all, an interesting story in which a student is the hero and outwits his teacher. That in itself will probably interest students, and perhaps they will remember it. Being fixed in their memory, the story can serve as a peg on which a mathematical idea can hang (in this case the method for summing arithmetic progressions). Like most biographical comments, t he story also reminds students t hat there are real people behind t he mathematics that they learn, that someone had to discover the formulas and come up with the ideas. Finally, especially when told as above, the story can lead the class towards discovering the formula for themselves.
But this example also raises some questions. That story appears in many different sources, with all sorts of variations. The sum is sometimes another, more complicated, arithmetic progression. The foolishness of the teacher is sometimes accentuated by including elaborate accounts of his reaction to Gauss's display of attitude. Many, but not all, versions include an account of Gauss's method. Such variations raise doubts about the story. Did it really happen? How do we know? Does it matter?
To some extent, it doesn't matter, but one might feel a little queasy telling students something that might not quite be true. In the case of our example, it's actually not hard to settle at least some of these quest ions. The story was told to his friends by Gauss himself when he was older. There is no particular reason to doubt its truth, though it's possible that it grew in t he telling, as stories that old men tell about themselves often do. The original version seems to have mentioned an unspecified arithmetic progression t h at involved much larger numbers, but overall the account above is likely not too far off the mark. Unfortunately, it is not always easy to find out whether an anecdote is true. So, when using an anecdote, ifs probably a good idea to make some sort of verbal gesture to suggest to students that what they are hearing may not necessarily be the strict historical truth.
The main limitation of using historical and biographical anecdotes. however, is that too often t hey are only distantly connected to the mathematics. This book, while including some such stories, hopes to point you toward some other ways of using history in the classroom, ways that more tightly intertwine the history with the mathematics.
One way to do t his is to use history to provide a broad overview. It is all too common for students to experience school mathematics as a random collection of unrelated bits of information. But that is not how mathematics actually gets created. People do things for a reason, and their work typically builds on previous work in a vast cross-generational collaboration. Historical information often allows us to share t his "big picture" with students. It also often serves to explain why certain ideas were developed. For example, Sketch 17, on complex numbers, explains why mathematicians were led to invent this new kind of number that initially seems so strange to students.
Most mathematics arises from trying to solve problems. Often the crucial insights come from crossing boundaries and making connections between subjects. Part of the "big picture" is the very fact that these links between different parts of mathematics exist. Paying attention to history is a way of being aware of these links, and using history in class can help students become aware of them.
History often helps by adding context. Mathematics, after all, is a cultural product. It is created by people in a particular time and place, and it is often affected by that context. Knowing more about this helps us understand how mathematics fits in with other hum an activities. The idea that numbers originally may have been developed to allow governments to keep track of data such as food production may not help us learn arithmetic, but it does embed arithmetic in a meaningful context right from the beginning. It also makes us think of t he roles mathematics still plays in running governments. Collecting statistical data, for example, is something that governments still do!
Knowing t he history of a n idea can often lead to deeper understanding, both for us and for our students. Consider, for example, the history of negative numbers (see Sketch 5 for t he details). For a long time after the basic ideas about negative numbers were discovered. mathematicians still found them difficult to deal with. The problem was not so much that they didn't understand the formal rules for how to operate with such numbers: rather, they had trouble with the concept itself and with how to interpret those formal rules in a meaningful way. Understanding this helps us understand (and empathize with) the difficulties students might have. Knowing how these difficulties were resolved historically can also point out a way to help students overcome these roadblocks for themselves.
History is also a good source of student activities. It can be as simple as asking students to research the life of a mathematician, or as elaborate as a project that seeks to lead students to reconstruct t he historical path that led to a mathematical breakthrough. At times, it can involve having (older) students try to read original sources. These are all ways of increasing student ownership of the mathematics by getting them actively involved.
In this book we have tried to supply you with material for all these ways of using history. The next part. "The History of Mathematics in a Large Nutshell," provides a concise overview of mathematic al history from earliest times to t he start of the 21st century and establishes a chronological and geographical framework for individual events. The thirty Sketches open up a deeper level of understanding of both the mathematics and the historical context of each topic covered. Finally, "What to Read Next" and t he bibliographical remarks sprinkled throughout the book suggest a vast array of resources for you or your students to use in pursuing further information about any of the ideas, people, or events that interest you.
Of course, there's much more to be said about how history may play a role in the mathematics classroom. In fact, this was the subject of a study sponsored by the International Commission on Mathematical Instruction (ICMI). The results of the study were published in. This is not an easy read, but it contains lots of interesting ideas and information. Many recent books combine history with pedagogy: some are collections of articles, such as. There is also an international society called known as whose full name is International Study Group on the Relations Between History and Pedagogy of Mathematics. The American section. HPM-Americas, runs regular meetings where both history and its use in teaching are explored.
In its journal Mathematics Teacher, the NCTM often publishes historical articles that contain ideas about how that history can be used in the classroom. Several packages of classroom-ready modules have been published, including and our own.
CHAPTER 2
The History of Mathematics in a Large Nutshell
The story of mathematics spans several thousand years. It begins as far back as the invention of the alphabet, and new chapters are still being added today. This overview should be thought of as a brief survey of that huge territory. Its intention is to give you a general feel for the lay of the land and perhaps to help you become familiar with the more significant landmarks.
Much (but by no means all) of the mathematics we now learn in school is actually quite old. It belongs to a tradition that began in the Ancient Near East, then developed and grew in Ancient Greece, India, and the medieval Islamic Empire. Later this tradition found a home in lat Medieval and Renaissance Europe, and eventually became mathematics as it is now understood throughout the world. While we do not entirely ignore other traditions (Chinese, for example), they receive less attention because they have had much less direct influence on the mathematics that we now teach.
Our survey spends far more time on ancient mathematics than it does on recent work. In a way, this is a real imbalance. The last few centuries have been times of great progress in mathematics. Much of this newer work, however, deals wit h topics far beyond the school mathematics curriculum. We have chosen, rather, to pay most attention to the story of those parts of mathematics that we teach and learn in school. Thus, the survey gets thinner as we come closer to the present. On the other hand, many of the topics we might have mentioned there appear in the sketches that make up t he rest of this book.
The study of the history of mathematics, like all historical investigation, is based on sources. These are mostly written documents, but sometimes artifacts are also important. When these sources are abundant, we are reasonably confident about the picture of the period in question. When they are scarce, we are much less sure. In addition, mathematicians have been writing about the story of their subject for many centuries. That has sometimes led to "standard stories" about certain events. These stories are mostly true, but sometimes historical research has changed our view of what happened. And sometimes historians are still arguing about the right story. In order to stay short, this survey ignores many of these subtleties. To make up for this, we provide references where you can find more information. To help you on your way, we have also provided an annotated list of books that might be good points of entry for further study. (See "What to Read Next," starting on page 223.)
As you read through this overview, you may be struck by how few women are mentioned. Before the 20th century, most cultures of Western civilization denied women access to significant formal education, particularly in the sciences. However, even when a woman succeeded in learning enough mathematics to make an original contribution to the field, she often had a very hard time getting recognized. Her work sometimes ended up being published anonymously, or by another (male) mathematician who had access to the standard outlets for mathematical publication. Sometimes it wasn't published at all. Only in recent years have historians begun to uncover the full extent of these obscured mathematical achievements of women.
In our times, most of the barriers to women in the sciences have been dissolved. Unfortunately, some of t he effects of the old "uneven playing field" still persist. The perception that mathematics is a male domain has been a remarkably resilient self-fulfilling prophecy. But things a re changing. The results of careful historical research and the outstanding achievements of many 20th century female mathematicians show that women can be creative mathematicians, have made substantial contributions to mathematics in the past, and will certainly continue to do so in the future.
Beginnings
No one quite knows when and how mathematics began. What we do know is that in every civilization that developed writing we also find evidence for some level of mathematical knowledge. James for numbers and shapes and the basic ideas about counting and arithmetical operations seem to be part of the common heritage of humanity everywhere. Anthropologists have found many prehistoric artifacts that can, perhaps, be interpreted as mathematical. The oldest such artifacts were found in Africa and date as far back as 37,000 years. They show that men and women have been engaging in mathematical activities for a long time. Modern anthropologists and students of ethnomathematics also observe that many cultures around the world show a deep awareness of form and quantity and can often do quite sophisticated and difficult things that require some mathematical understanding. These range all the way from laying out a rectangular base for a building to devising intricate patterns and designs in weaving, basketry, and other crafts. These mathematical (or pre-mathematical) elements of current preliterate societies may be our best hint at what the earliest human mathematical activity was like.
By about 5000 B.C., when writing was first developing in the Ancient ear East, mathematics began to emerge as a specific activity. As societies adopted various forms of centralized government, they needed ways of keeping track of what was produced, how much was owed in taxes, and so on. It became important to know the size of fields, the volume of baskets, the number of workers needed for a particular task. Units of measure, which had sprung up in a haphazard way, created many conversion problems that sometimes involved difficult arithmetic. Inheritance laws also created interesting mathematical problems. Dealing with all of these issues was the specialty of the "scribes." These were usually professional civil servants who could write and solve simple mathematical problems. Mathematics as a subject was born in the scribal traditions and the scribal schools.
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Excerpted from "Math Through the Ages"
by .
Copyright © 2014 William P. Berlinghoff and Fernando Q. Gouvêa.
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