**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.*

Reprinted with permission from [name of book or
journal], copyright [December, 1987] by the National Council of Teachers of
Mathematics. All rights reserved.

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.