Using Problems from the History of Mathematics

Introduction

Many teachers believe that the history of mathematics, if incorporated into school lessons, can do much to enrich its teaching. If this enrichment is just the inclusion of more factual knowledge in an already crowded curriculum, the utility and appeal of historical materials for the classroom teacher is limited. Thus, to include a historical note in a student's text on the life or work of a particular mathematician may shed a historical perspective on the content, but does it actually encourage learning or illuminate the concept being taught? The benefits of this practice can be debated.

A more direct approach to historically enriching mathematics instruction and the learning of mathematics is to have students solve some of the problems that interested early mathematicians. Such problems offer case studies of many contemporary topics encountered by students in class. They transport the reader back to the age when the problems were posed and illustrate the mathematical concerns of the period. Often, these same concerns occupy modern-day mathematics students. This simple realization, namely, the continuity of mathematical concepts and processes over past centuries, can help motivate learning. Students can experience a certain thrill and satisfaction in solving problems that originated centuries ago. In a sense, these problems allow the students to touch the past.

Strategies for Employing Historical Problems

In a 1985 issue of the Mathematics Teacher (Wikenfeld), a problem included in “Reader Reflections” prompted a series of follow-up discussions in later issues of the journal. Obviously, the problem caught the attention of the readers. Interesting mathematical variants were proposed and solved (Lieske, 1985). The problem was good because it was strikingly simple in its conception: ”Given a right triangle with legs of length a and b and the hypotenuse of length c, what is the length of a side of the largest inscribed square having the right angle as one of its vertices?” See the figure below.

Interestingly, the side of the square is found to be the product of the legs divided by the sum of the legs. This problem takes on even more intrigue when one learns that it was first known to be posed over 2000 years ago in China. It is the fifteenth problem in the ninth chapter of Jiuzhang suanshu (Nine chapters on the mathematical art, Swetz and Kao, 1977). Now, if this problem is examined in its historical context, it could be noted that Problem 16 in this collection challenged the reader to find the radius, r, of the inscribed circle for a given right triangle.

In a similar manner, a present-day teacher can offer this challenge to a first-year algebra class. Can they find the same answer for this problem that the ancient Chinese scribes found; namely, that r equals the product of the legs divided by the sum of the three sides? The mathematical learning experience built around these problems can then be further extended by an examination and discussion of the Chinese geometric-algebraic solution scheme for the inscribed-circle problem as shown below:

For historical interest, a facsimile of the actual Chinese solution's illustration is presented, including a graphical error in the circle's construction. Reference lettering is a modern insertion; however, the use of a grid is a traditional Chinese procedure.

Several problems in this same series are  ingenious in their conception and require true perceptual and mathematical acuity on the part of the solver. To form a solution strategy, the use of a diagram is more than a suggestion; it is imperative. Problems 5 and 9 in the series are examples of this type:

• A tree of height 20 feet has a circumference of 3 feet. There is an arrowroot vine which winds seven times around the tree and reaches the top. What is the length of the vine? [Answer: 29 ft.]
• A wooden log is encased in a wall. If we cut part of the wall away, at a depth of 1 inch, the width of the exposed log measures 1 foot. What is the diameter of the log? [Answer: 37 in.]

Assembling Thematic Sequences of Problems

Historical sequences of problems from different time periods and cultures can be assembled and assigned as exercises for students to solve and compare. For example, facility in the use of the Pythagorean theorem was valued in all societies:

• A beam of length 30 feet stands against a wall. The upper end has slipped down a distance of 6 feet. How far did the lower end move? (Babylonia, 1800-1600 B.C.E.). [Answer: 18 ft.]
• The height of a wall is 10 feet. A pole of unknown length leans against the wall so that its top is even with the top of the wall. If the bottom of the pole is moved 1 foot farther from the wall, the pole will fall to the ground. What is the length of the pole? (China, 300 B.C.E.) [Answer: 50.5 ft.]

More mathematically mature societies puzzled over indeterminate equations:

• 576 coins have been paid for the purchase of 78 bamboo poles. It is desired to calculate prices for large and small poles. How much is the price of each? (China, C.E. 300) [An answer: 48 small poles at 7 coins each; 30 large poles at 8 coins each]
• A hundred bushels of grain are distributed among 100 persons in such a way that each man received 3 bushels, each woman 2 bushels, and each child half a bushel. How many men, women, and children are there? (Europe, 775) [An answer: 20 men, 0 women, and 80 children]

At times, in instructional situations, the use of historical problems reinforces and clarifies the concept being taught. For example, a discussion on the technique of "completing the square" to find the roots of a quadratic equation is enhanced by reference to actual Babylonian problems from which the concept originated (McMillan, 1984). Consider a problem from 2000 B.C.E.:

If the side of the square in question is taken to be x, then the problem becomes one of solving the equation

x2 + (2/3)x = 35/60

which in Babylonian methodology would be depicted as shown here:

Thus by geometrically completing the square, the new area (see below) is seen to be 35/60 + 1/9 and the algebraic solution situation becomes one of solving

(x + 1/3)2 = 35/60 + 1/9.

Assisted by the use of a hand calculator, modern students would find that x = 1/2 or x = -7/6, and the positive root agrees with the solution found by the Babylonians.

Students as Mathematical Archaeologists

Students can be placed in the role of mathematical archaeologists and led to discoveries. Assume that the class has discussed numeration systems including the sexagesimal Babylonian cuneiform system. They are presented with a facsimile of the face of a clay tablet from the Yale Babylonian Collection (7289):

If recognized as numerals, the symbols along the horizontal diagonal can be translated as 1, 24, 51, 10. Babylonian numerals possess a positional value. Thus, if the 1 is interpreted as one unit, the remaining numerals represent fractional components of the number represented, namely, 24/60, 51/602 and 10/603  When the numbers are combined, the mystery number is found to be 1.414 212 9.... It approximates √2 to six decimal places--impressive accuracy for 2000 B.C.E.! Two discoveries emanate from this example: the ancient Babylonians performed geometric constructions similar to those in use today, and they had a proficient technique for the extraction of square roots. Babylonian accuracy in the extraction of square roots prompts further investigation by students--just how did they do it?

Cultural and sociological information can be obtained from the solutions to historical problems. For example, the height of the mast of an Egyptian ship for the period 250 B.C.E. can be found:

• If it is said to you, “Have a sailcloth made for the ships” and it is further said, "Allow 1000 cloth cubits (square cubits) for one sail and have the ratio of the height of the sail to its width as 1 to 1 1/2." What is the height of the sail? (1 cubit = 20 inches) [Answer: 25.8 cubits]

Similarly, the size of a loaf of bread in fifteenth-century Venice can be deduced:

• When a bushel of wheat is worth 8 lire, the bakers make a loaf of bread weighing 6 ounces; required the number of ounces in the weight of a loaf when wheat is worth 5 lire a bushel. [Answer: 9 3/5 ounces]

Or the hourly wages (of a twelve-hour workday) for a man in post-Civil War America can be determined:

• A gentleman received $4 a day for his labor, and pays$8 a week for his board; at the expiration of 10 weeks he has saved \$144; required the number of idle and working days. [Answer: 14 idle days and 56 working days]

Problems Representing Transitions in Mathematical Thought

Sets of study problems that demonstrate a transition in mathematical thought can be given to students. Mathematical problems from early periods in a society are usually empirically based and task oriented:  taxes had to be computed, dikes and walls constructed, grain stored, and the price of merchandise reckoned. Later in the history of the society, one often finds the same problem situations modified to promote mathematical techniques. Thus, in China in 300 B.C.E., surveying concerns prompted the formulation and solution of problems involving towns and landmarks. Typical of such problems is that of a square-walled town. Several measurements outside the walls are given, and the width of the town is desired:

The computational situation gives rise to a quadratic equation where x is the desired width of the town’s wall:

x 2 + 34x - 71000 = 0,

for which the root x = 250 is obtained. By the thirteenth century, such problems had passed out of the realm of surveyors and had become the basis for rather powerful algebraic demonstrations.

An analysis of the Chinese solution processes for the round-wall situation described in the figure below indicates the solution of a tenth-degree equation,

x10 + 15x8 + 72x 6 - 864x 4- 11 664x 2 - 34992 = 0.

A root of x = 3 was found for the solution. The problem's author, Qin Jiushao, however, was showing off and exhibiting much mathematical bravado. If one lets the diameter of the town be x instead of x2, a much simpler equation for the problem can be found:

x 4 + 6x3 + 9x2 - 972x - 2916 = 0.

Similar situations also abound in European mathematical literature. Problems of partnership were a primary concern in Renaissance Italy of the fifteenth century:

• Three men, Tomasso, Domengo and Nicolo, entered into partnership. Tommaso put in 760 ducats on the first day of January, 1472, and on the first day of April took out 200 ducats. Domengo put in 616 ducats on the first day of February, 1472, and on the first day of June took out 96 ducats. Nicolo put in 892 ducats on the first day of February, 1472, and on the first day of March took out 252 ducats. On the first day of January, 1475, they found that they had gained 3168 ducats, 13 grossi and 1/2. Required the share of each, so that no one shall be cheated. (1 ducat = 24 grossi) [Answer: Tomasso, 1061 ducats, lg; Domengo 949 ducats, 19 1/2 g; Nicolo 1157 ducats, 17g]

This problem can be solved by the use of simple proportions, but a century later partnership problems served as a basis for complex algebraic computation:

• Four men form an organization. The first deposits a given quantity of aurei; the second deposits the fourth power of one-tenth of the first; the third, five times the square of one-tenth the first; and the fourth, 5. Let the sum of the first and second equal the sum of the third and fourth. How much did each deposit? [Answer: 7.4907, 0.3148, 2.8055, and 5, respectively]

Of course problems with historical allure can be assigned merely to demonstrate the mathematical capabilities and skill of our predecessors. A Susa tablet of 2000 B.C.E. requires the reader to find the radius of a circle in which is inscribed an isosceles triangle of sides 50, 50, and 60. An Egyptian counterpart of 150 C.E. asks the reader to find the area of a circle that circumscribes an equilateral triangle whose side is 12 units:

In its Babylonian context, this problem is solved by computing the length of the altitude drawn to the 60-unit side. (The length of the altitude is 40 units.) Radii, r , are then visualized extending to the three vertices of the triangle, and the relationship is established.

r 2 = 302 + (40 - r)2

Radius r is found to be 31.25 units. The scribe solving the second-century Egyptian counterpart to this problem first used the Pythagorean proposition to find the altitude of the triangle, √108 units, and then applied this result to obtain the area of the triangle, 6√108 square units. Next, he employed a trapezoidal approximation for the area of the remaining circular segment.  Multiplying the obtained area by 3, since three congruent segments were involved, he added the result to the triangle's area to obtain a total area for the circle. It appears that the Egyptian calculations were performed using the Babylonian sexagesimal system and the resulting sexagesimal fractions converted into the Egyptian format of sums of unit fractions. The scribe's computation resulted in an answer of 143 + 1/10 + 1/20. How accurate is this result? Is the scribe’s confidence in the trapezoidal approximation for a circular segment well founded?

Conclusion

When visiting a large art museum one commonly finds groups of schoolchildren accompanied by their teachers admiring and studying paintings, sculptures, and other works of art from centuries past. Their teachers or guides draw the students' attention to the artist's techniques: mastery of color, tone, interplay of light and shadow, and even the significance of the scenes described. Under this directed scrutiny, a painting or statue becomes a testimony to its creator's genius and offers some understanding of the period in which the artist lived and functioned. Learning takes place. This learning is both cognitive and affective. So, too, are the mathematics problems of history. They, in a sense, are intellectual and pedagogical works of art that testify to an expression of human genius. But, unlike the museum pieces, these problems can actually be possessed by the viewers through participation in the solution processes. Questions originating hundreds or even thousands of years ago can be understood and answered in today's classrooms. What a dramatic realization that is!

Historical problems and problem solving, as a topic in itself, can be the focus of a lesson, but such problems are probably more effectively interspersed among classroom drills and homework assignments. Teachers who like to specify a “problem of the week” will find that historical problems fit the task nicely. Ample supplies of historical problems can be found in survey books on the history of mathematics or in the Convergence feature, Problems from Another Time. The seeking out and employing of historical mathematical problems in classroom instruction is a rewarding and enriching experience in which mathematics teachers should partake.

References

Lieske, G. Spencer. “Reader Reflections: Right Triangle,” Mathematics Teacher 78 (October 1985): 498-99.

McMillan, Robert. “Babylonian Quadratics,” Mathematics Teacher 77 (January 1984): 63-65.

Swetz, Frank J. (editor), "Problems from Another Time," Convergence (2004).
http://mathdl.maa.org/mathDL/46/?pa=content&sa=browseNode&browseTypeId=12

Swetz, Frank J. and T. I. Kao. Was Pythagoras Chinese? An Examination of Right Triangle Theory in Ancient China. Reston, VA: National Council of Teachers of Mathematics, 1977.

Wikenfeld, Morris. “Reader Reflections: Right Triangle Relationships,” Mathematics Teacher 78 (January 1985): 12.

Author's Note: This article is a revised version of an article that appeared in Mathematics Teacher 82 (May 1989), pp. 370-377, and it appears in Convergence by permission of the National Council of Teachers of Mathematics.