Using Historical Materials

Using Historical Materials

in the Mathematics Classroom

By Abraham Arcavi

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

The arguments advocating the use of the history of mathematics in mathematics education have become widespread in recent years. Theoretical and practical guidelines for using history have also appeared, some of them accompanied by the description of actual experiences.

The present article, which is intended to promote the use of history in the mathematics classroom, describes and discusses an activity organized 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 Englishman 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 copied 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 various topics including arithmetic, the calculation of ' areas, and the resolution of "linear equations.”

In the following activity we concentrate on two extracts dealing with arithmetical operations.

The activity

The activity can be introduced by a historical account similar to the foregoing, to set the scene and motivate the Students. In the following a description of Egyptian writing is presented. It should be noted that ancient Egyptian writing had two forms, hieroglyphic and hieratic. Hieroglyphics are mainly to be found as inscriptions on stone in temples and sepulchres. Hieratic writing is a cursive script, quicker to write, used mainly in the papyri. In figure 1 we can see some of the hieroglyphic number symbols, which are easier to decipher 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 versions 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 opportunity 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 symbol represents ten identical lesser symbols;

* numbers are formed by juxtaposition of symbols, but no place value is used, that is, if the symbols designating a number are rearranged, 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 students' 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

I X 2801 + 2 x 2801 + 4 x 2801 = 19607

And if we rewrite it using the distributive law, we obtain

(1+ 2 + 4) x 2801 = 19607

which means that the arithmetic operation 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 calculation method of the Egyptians in this problem.

The next exercise is intended to reinforce students' discovery in a similar situation, with a slight variation. Again the students are asked to rewrite 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 something 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 approach the next step.

Exercise 3: Calculate 13 X 27 by the Egyptian method

The objective of this question is twofold. First, the students are asked to practice by themselves a multiplication by "doubling and summing up with different numbers. Once they realize that the calculation can be performed 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 illustration of the commutative law of multiplication.

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 randomly chosen factors could prompt the discussion.

Exercise 4: Can one multiply any pair of numbers by the Egyptian method? Explain

This upper-level mathematical question 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 differently by different students. Some (if not most) of them will make many trials and then "jump" to a conclusion, which, if correct, is indeed acceptable. 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 answer still cannot be certain unless we find a general justification if the answer is affirmative (or a counter example 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 general 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 formulations in the source

* Redoing the mathematics in modern notation. leading to an understanding of what was done

* Applying the operation or process to other examples

* Discussing the mathematics involved with our hindsight (justifications, generalizations, etc.)

Many primary sources supply a rich mathematical environment for such activities, especially when one wants to review and deepen the understanding 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 analysis 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 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 principles of binary arithmetic.” No evidence 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 foregoing will depend only on the historian's interpretation. We must be aware that when we are looking at what the Egyptians have done with our experience 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 ancient Egyptian mathematics, contigual extracts should be read by the teacher that uses the activity proposed here.

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

References

Booker, G. Review or the HPM meeting at 1CME 5. In Jones, C. V., ed., Newsletter of the International Study- Group on the Relations between History and Pedagogy of Mathematics 8, pp. 54. 1985. (Available free from the editor, Department of Mathematical Sciences, Ball State University. Muncie, IN 47306)

Chace. A. B. The Rhind Mathematical Papyrus. 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 Mathematics. Englewood Cliffs, N.J.: Prentice-Hall, 1976.

Kreitz. H. M.. and F. Flournoy. "A Bibliography of Historical Materials for Use in Arithmetic in the Intermediate Grades.- Arithmetic Teacher 7 (1960):287-92.

May. K. 0. 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 Publishers, 1975.

Read, C. B., and J. K. Bidwell. “Selected Articles Dealing with the History of Elementary Mathematics,” School Science and Mathematics 76 (1976):477-83.