using historical materials in the mathematics classroom

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Using Historical Materials in the Mathematics Classroom Author(s): Abraham Arcavi Source: The Arithmetic Teacher, Vol. 35, No. 4 (December 1987), pp. 13-16 Published by: National Council of Teachers of Mathematics Stable URL: http://www.jstor.org/stable/41194281 . Accessed: 12/06/2014 12:39 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to The Arithmetic Teacher. http://www.jstor.org This content downloaded from 62.122.79.31 on Thu, 12 Jun 2014 12:39:44 PM All use subject to JSTOR Terms and Conditions

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Page 1: Using Historical Materials in the Mathematics Classroom

Using Historical Materials in the Mathematics ClassroomAuthor(s): Abraham ArcaviSource: The Arithmetic Teacher, Vol. 35, No. 4 (December 1987), pp. 13-16Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41194281 .

Accessed: 12/06/2014 12:39

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Arithmetic Teacher.

http://www.jstor.org

This content downloaded from 62.122.79.31 on Thu, 12 Jun 2014 12:39:44 PMAll use subject to JSTOR Terms and Conditions

Page 2: Using Historical Materials in the Mathematics Classroom

Using Historical Materials in the Mathematics Classroom

By Abraham Arcavi

The arguments advocating the use of the history of mathematics in math- ematics education have become wide- spread in recent years. Theoretical and practical guidelines for using his- tory have also appeared, some of them accompanied by the description of actual experiences (see, e.g., the review by Booker [1985]).

The present article, which is in- tended to promote the use of history in the mathematics classroom, de- scribes and discusses an activity orga- nized around a primary historical source. Also a general framework is proposed that can be used for the creation of similar activities based on other mathematical topics to be found in other historical sources.

The Source: The Rhind Mathematical Papyrus The Rhind Mathematical Papyrus is one of the oldest extant mathematical documents. The papyrus takes its name from Henry Rhind, an English-

Abraham Arcavi teaches at Ball State Univer- sity, Muncie, IN 47306. His present activities are related to the use of the history of mathe- matics in mathematics education.

man who bought it in Luxor, Egypt, in 1858. After his death it came into the possession of the British Museum, where it remains today. The papyrus is also associated with the name of Ahmes, the Egyptian scribe who cop- ied it. It is estimated that the papyrus dates from the seventeenth century b.c. but is, apparently, a copy of even earlier sources.

The papyrus contains a collection of eighty-seven problems and their solutions. The problems cover vari- ous topics including arithmetic, the calculation of areas, and the resolu- tion of "linear equations."

In the following activity we concen-

trate on two extracts dealing with ar- ithmetical operations. The activity The activity can be introduced by a historical account similar to the fore- going, to set the scene and motivate the students. In the following a de- scription of Egyptian writing is pre- sented. It should be noted that ancient Egyptian writing had two forms, hi- eroglyphic and hieratic. Hieroglyph- ics are mainly to be found as inscrip- tions on stone in temples and sepulchres. Hieratic writing is a cur- sive script, quicker to write, used mainly in the papyri. In figure 1 we

Fig. 1

1=j 1 ,000 = T (lotus flower)

10= H 10,000= j| (bent finger)

100 = C) 100,000 = fcf (tadpole)

1 ,000,000 = Y-£ (man with raised arms)

December 1987 13

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Page 3: Using Historical Materials in the Mathematics Classroom

Fig. 2

From the solution to Problem 79.

Hieratic Hieroglyphic Modern

| 9999 ? V 1 2801

tfl^íui II 999????? 2

"'999Ш^ TOlal 19607

Sii

can see some of the hieroglyphic num- ber symbols, which are easier to deci- pher than in hieratic script.

After this introduction, the students can begin reading the extracts chosen (taken from the solutions to larger problems in the papyrus) with the help of guiding exercises and questions. The hieratic and the hieroglyphic ver- sions of the extracts are taken from Chace (1969) and Peet (1970), respectively.

Exercise 1: Complete the blanks in the "Modern" column in figure 2.

The students should be supplied with a copy of figure 1 , which enables them to rewrite the numbers in "modern" (Hindu- Arabic) numerals, so that they can undertake the first step toward deciphering the text.

The exercise also lends an opportu- nity to discuss some properties of a different numeration system and to compare its characteristics to ours. For instance, it should be noted that in the Egyptian system -

• some "decimal" characteristics appear, in the sense that one sym- bol represents ten identical lesser symbols;

• numbers are formed by juxtaposi- tion of symbols, but no place

value is used, that is, if the sym- bols designating a number are re- arranged, they still represent the same number;

• no symbol for zero occurs (there is no need for it because Egyptian numeration does not have place value).

Exercise 2: What is the calculation being done? And what is the method?

In this step the students are required to understand what mathematics is being "done" in the extract. Looking at the completed "modern" column, we see the following:

1 2801 2 5602 4 11204

Total 19607

Probably, the first thing to observe is that, a sum was performed: 2801 + 5602 + 11204 = 19607. Then the stu- dents' attention can be directed toward the numbers 1,2, and 4 and their roles.

Each of these numbers (except 1) is the double of the preceding; then it is observed that the same is true of 2801 , 5602, and 11204. This observation ¡should lead to the realization that the operation performed is

1x2801 + 2x2801 + 4x2801 = 19607.

And if we rewrite it using the distrib- utive law, we obtain

(1 + 2 + 4)x2801 = 19607, which means that the arithmetic oper- ation performed is none other than 7x2801 = 19607. The students, using their observation and arithmetical knowledge, are led to decipher, in a guided-discovery process, the calcu- lation method of the Egyptians in this problem.

The next exercise is intended to reinforce students' discovery in a sim- ilar situation, with a slight variation. Again the students are asked to re- write the text in modern notation and to decipher the operation performed. In this problem, if they proceed in an analogous manner, they obtain the following:

/ 1 2000 2 4000

/ 4 8000 Total 10000

At this point, they perceive that some- thing is "wrong." The hint to be given is to pay attention to the slash marks (/) at the side of certain numbers, to realize that not all of them should be added, but only those marked.

Then the students are ready to ap-

14 Arithmetic Teacher

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Page 4: Using Historical Materials in the Mathematics Classroom

Fig. 3

From the solution to Problem 52.

Hieratic Hieroglyphic Modern

¿•' ffîî " 2 - «il __' klik IMI/ /4

(S w Total 10000

December 1987 15

proach the next step.

Exercise 3: Calculate 13 x 27 /ry ř/zď Egyptian method

The objective of this question is two- fold. First, the students are asked to practice by themselves a multiplica- tion by "doubling and summing up," with différent numbers. Once they re- alize that the calculation can be per- formed in one of the two following ways,

/ 1 27 / 1 13 2 54 / 2 26

/ 4 108 4 52 / 8 216 /8 104

Total 351 /16 208 Total 351

it is desirable to encourage them to do both not only to practice the method twice but also to see a further illustra- tion of the commutative law of multi- plication.

The second purpose is to prepare the students for the next exercise. The fact that the preceding proposed a multiplication with two "ugly" (one odd, one prime) and apparently ran- domly chosen factors could prompt the discussion.

Exercise 4: Can one multiply any pair of numbers by the Egyptian method? Explain This upper-level mathematical ques-

tion is designed to induce the student to think mathematically, that is, to investigate and to generalize from a particular situation already learned and understood.

The answer will be approached dif- ferently by différent students. Some (if not most) of them will make many trials and then "jump" to a conclu- sion, which, if correct, is indeed ac- ceptable. At this point, and without further sophistication, it is advisable (if the level of the class allows it) to introduce the idea that no matter how many cases one can check, the an- swer still cannot be certain unless we find a general justification if the an- swer is affirmative (or a counter exam- ple otherwise).

The answer to exercise 4 is the same as the answer to the following: can any number be written as the sum of powers of two? Yes! This fact is the basis of binary arithmetic! Thus the Egyptian method of multiplication works for any choice of integers.

The Framework The foregoing is an example of a gen- eral framework for an activity that can be developed (at any mathematical level) around a primary source. The framework includes the following steps:

• "Dictionary" questions that help one to become acquainted with

unknown notations, symbols, names of concepts, or formula- tions in the source

• Redoing the mathematics in mod- ern notation, leading to an under- standing of what was done

• Applying the operation or process to other examples

• Discussing the mathematics in- volved with our hindsight (justifi- cations, generalizations, etc.)

Many primary sources supply a rich mathematical environment for such activities, especially when one wants to review and deepen the understand- ing of a topic already learned, without provoking a deja vu feeling, as might be the case of our example.

Furthermore, the historical context may motivate the student and also can be a way of connecting mathematics to other subjects.

Last But Not Least: The Historian's Point of View Our main interest in primary historical sources is pedagogical. Nevertheless a cautionary word may be said from the historical point of view. The anal- ysis and interpretation of historical documents is not a straightforward subject. Historians usually differ in the way they look at the same source. Thus, for instance, our interest as

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Page 5: Using Historical Materials in the Mathematics Classroom

teachers on raising mathematical questions from the source, such as the generality of the Egyptian method for multiplication, should not lead us to careless historical conclusions like "the Egyptians knew the basic princi- ples of binary arithmetic." No evi- dence in the Papyrus could suggest that they were concerned at all about the generality of their method. So, the truth of statements such as the fore- going will depend only on the histori- an's interpretation. We must be aware that when we are looking at what the Egyptians have done with our experi- ence and hindsight, the boundaries between their knowledge and ours may be blurred.

Also, we have to be aware that the extracts have been looked at out of the context of the whole source. To have a more complete picture of an- cient Egyptian mathematics, contig- ual extracts should be read by the teacher that uses the activity pro- posed here.

The historical caveat notwithstand- ing, primary sources offer a bountiful and as yet unexploited supply of mathematical learning activities.

References

Booker, G. Review of the HPM meeting at ICME 5. In Jones, C. V., ed., Newsletter of the International Study Group on the Rela- tions between History and Pedagogy of Mathematics 8, pp. 5-8. 1985. (Available free from the editor, Department of Mathematical Sciences, Ball State University, Muncie, IN 47306)

Chace, A. B. The Rhind Mathematical Papy- rus. Reston, Va.: NCTM, 1969.

Peet, T. E. The Rhind Mathematical Papyrus. Liverpool: University of Liverpool Press, 1970.

Bibliography Bunt, L. N. H., P. S. Jones, and J. D. Bedient.

The Historical Roots of Elementary Mathe- matics. Englewood Cliffs, N.J.: Prentice- Hall, 1976.

Kreitz, H. M., and F. Flournoy. "A Bibliogra- phy of Historical Materials for Use in Arith- metic in the Intermediate Grades." Arith- metic Teacher 7 (1960): 287-92.

May, K. O. Bibliography and Research Manual of the History of Mathematics. Toronto: Toronto University Press, 1973.

Popp, W. History of Mathematics: Topics for Schools. Translated from the German by M. Bruckheimer. London: Transworld Publish- ers, 1975.

Read, С. В., and J. K. Bidwell. "Selected Articles Dealing with the History of Elemen- tary Mathematics." School Science and Mathematics 76 (1976): 477-83. W

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