Women and Minorities in Mathematics

Incorporating Their Mathematical Achievements Into School Classrooms

 

 

Thomas Fuller and his Calculation Ability

 

 

Sarah J. Greenwald

Appalachian State University

Boone, North Carolina

 

Amy Ksir

United States Naval Academy

Annapolis, Maryland

 

Lawrence H. Shirley

Towson University

Towson, Maryland

 

 

In 1792, Thomas Jefferson, who would later become the third president of the United States, said:

 

Comparing them by their faculties of memory, reason, and imagination, it appears to me that in memory [the Negro] are equal to the whites; in reason much inferior, as I think one could scarcely be found capable of tracing and comprehending the investigations of Euclid; and that in imagination they are dull, tasteless, and anomalous. (Williams, 1999d)

 

There are many counterexamples to Jefferson’s claims.  The existence of African American mathematicians before and during the time of Thomas Jefferson, such as Benjamin Banneker and Thomas Fuller, contradict Jefferson's assertions.  In fact Banneker, living in the U.S. at the time, responded to Jefferson’s comments with a twelve-page letter (Williams, 1999b).  Not only are people of African descent capable of studying and understanding known mathematics, but they have also shown imagination, creativity and mathematical ability in their investigations of original mathematics (Williams, 1999c).

 

There has been much written about Benjamin Banneker, including ideas for incorporating his mathematics into classrooms (e.g., Johnson, 1999; Lumpkin, 1997a; Lumpkin, 1997b; Lumpkin & Strong, 1995; Smith, 1996).  Less has been written about Thomas Fuller, another African American mathematician who lived during the same time period.  Thomas Fuller was a slave who possessed remarkable calculation abilities.  This article discusses Fuller, his mathematical ability, the ethnomathematical context, and related classroom activities.

 

Thomas Fuller

 

Thomas Fuller was born in Africa in 1710.  At the age of 14, he was sold into slavery and he was taken to Virginia.  While he never learned how to read or write, he had an amazing ability to perform mental calculations.  In 1788, abolitionists interviewed Fuller in order to demonstrate that African American men were not mentally inferior to white men.  They asked him to perform a number of calculations.  One of the questions was to compute the number of seconds a man who is 70 years, 17 days, and 12 hours old has lived.  Astoundingly, he answered 2,210,500,800 (the correct answer) in only a minute and a half.  According to a newspaper account:

 

One of the gentlemen, who employed himself with his pen in making these calculations, told him [Fuller] he was wrong, and that the sum was not so great as he had said—upon which the old man hastily replied, ‘top, massa, you forget de leap year.’  On adding the seconds of the leap years to the others, the amount of the whole in both their sums agreed exactly.”

 

In addition, when one of the men remarked that it was a shame that Fuller had never had a formal education, Fuller replied, “No, massa, it is best I got no learning; for many learned men be great fools.”  One should remember that Fuller’s style of speech was typical of accounts of slaves (usually written by whites) during this time.  In addition, it would have been unusual for an African American slave to correct or contradict a white man, and the fact that Fuller did so is worth noting.

 

Today no one knows exactly how Thomas Fuller performed his calculations.  However, the algorithms he used were probably based on traditional African counting systems.  The people of the Yoruba area of southwest Nigeria have a complex counting system with very high numbers that probably dates back to Fuller’s time.  Europeans arriving in the area were amazed at the complexity of Yoruba numeration.  It is thought to have developed from counting the cowrie shells that were used for currency.  Economic inflation may have raised the magnitude of the numbers to be counted.  Yoruba numeration has a well-organized structure, base twenty with an intermediate base ten, that allows for easy calculation and has provisions for large numbers as multiples and powers of twenty.  Yoruba also uses subtraction that is similar to the “IX” for nine in Roman numerals.  For example, the numbers from fifteen to nineteen are expressed as subtractions from twenty, the base number.  This may also help with calculation, since calculating with “twenty minus three” might be easier than dealing with seventeen.

 

We have additional evidence of superior calculation abilities on the coast of Benin from John Bardot’s 1732 account of the abilities of the inhabitants of Fida (Fauvel & Gerdes, 1990):

 

The Fidasians are so expert in keeping their accompts [accounts], that they easily reckon as exact, and as quick by memory, as we can do with pen and ink, though the sum amount to never so many thousands:  which very much facilitates the trade the Europeans have with them.

 

In 1788, Thomas Clarkson discussed the calculation ability of an African slave broker (Fauvel & Gerdes, 1990):

 

He reduces them immediately by the head to bars, coppers, ounces, according to the medium of exchange that prevails in the part of the country in which he resides, and immediately strikes the balance.  The European, on the other hand, takes his pen, and with great deliberation, and with all the advantages of arithmetick [sic] and letters, begins to estimate also.  He is so unfortunate often, as to make a mistake; but he no sooner errs, than he is detected by this man….  Incidences of this kind are very frequent.

 

The Bassari people of southeastern Senegal also have a mathematical tradition that dates back to Fuller’s time.  Recently, Ron Eglash talked to a Bassari elder about calculation ability (Eglash, 1999):

 

The Bassari elder who demonstrated these tallies to me… told me that he did not know much about traditional forms of calculation, but he did know that in pre-colonial times it was performed by specialists who were trained in memorization of sums.

 

Knowing which part of Africa Fuller came from might shed light on the kinds of algorithms he used.  While we hope that math historians will someday track the African heritage of Fuller, Alex Haley was an exception in his success at tracking individual slaves back to an area of origin, and so this would probably be very difficult.  However, some guesses can be made.  Eglash (1999) discusses some theories:

 

Curtin (1971) shows that the slave trade from what is now northern Senegal diminished after 1700, and that the Nigerian area did not begin major activity until after 1730. This still leaves the possibility that Fuller came from the area of present-day Benin and Ghana, which would be too far south to have directly shared influences with the Bassari, but Holloway (1990, pg 10) notes that Virginians showed some preference for Africans from the Senegambian region.

 

Even if we knew Fuller’s birthplace, the next problem would be to document more fully the mathematical thinking that was being done at the time.  This might be even more difficult given the paucity of historical records, especially about mathematical thinking.

 

Classroom Activities

 

Lumpkin & Strong (1995) present a number of classroom worksheets about Fuller.  We offer additional activities designed to introduce ideas related to Fuller and his calculation ability to students.  Classroom Activity Sheets 1 and 2 can be found at the end of this column along with select teacher solutions.

 

Introductory Activity

Introduce Fuller to the class and calculate of the number of seconds a man who is 70 years, 17 days, and 12 hours old has lived.  Tell the class that he answered 2,210,500,800 (the correct answer) in only a minute and a half.  Break the class up into two groups.  Allow one group to use only calculators in order to identify with Africans with superior mental calculation ability.  Allow the other group to use only a pen and paper.  Present problems to the class and time the groups.  Bring the class back together and relate the activity to the quote from Thomas Clarkson (see above) about the comparison of the mental calculation ability of an African slave broker with the pen and paper work of a (white) European.

 

Activity Sheet 1:  Yoruba Numeration and Calculation Algorithms

Students will study the patterns of Yoruba counting words and practice expressing numbers in the Yoruba language.  They will also see how the distributive law used with subtraction can assist with calculation.

 

Activity Sheet 2:  Calculation Time

Students will need a way to time themselves.  During the in-class portion, students will complete several computations similar to Fuller’s calculations.  They will compute mentally, on paper, and with a calculator, and they will record how long it takes.  They will then pair up with another student to compare answers.  There is also a take-home portion.  Students will do Web research on early computing machines and they will examine and write about ideas relating to computation speed.

 

NCTM Standards

Classroom activities relating to Thomas Fuller will address several parts of the NCTM Principles and Standards for School Mathematics.  The NCTM has standards for computational fluency, including mental computations and working with very large numbers, and the suggested activities should stretch students’ abilities in both areas.  The activities we suggest are also relevant to the NCTM standards on problem solving, asking students to develop their own problem solving strategies for these mental calculations, and on communicating mathematical ideas, by comparing their answers with another student and evaluating each others’ solutions.  One might also consider a discussion of Thomas Fuller’s life, and how his mathematical ability was used by abolitionists, as addressing the standard on connections between mathematics and other contexts.

 

References

 

Eglash, R. (1999). African fractals:  modern computing and indigenous design. New Brunswick, NJ: Rutgers University Press.

 

Fauvel, J. & Gerdes, P.  (1990). African slave and calculating prodigy: Bicentenary of the Death of Thomas Fuller.  Historia Mathematica, 17, 141-151.

 

Gerdes, P.  (1994). On mathematics in the history of sub-saharan Africa.  Historia Mathematica, 21, 345-376.

 

Johnson, A.  (1999).  Famous problems and their mathematicians (pp. 114-115).  Englewood, CO:  Teacher Ideas Press.

 

Lumpkin, B. (1997a).  Algebra activities from many cultures.  Portland, ME:  J. Weston Walch.

 

Lumpkin, B. (1997b).  Geometry activities from many cultures.  Portland, ME:  J. Weston Walch.

 

Lumpkin, B. & Strong, D.  (1995).  Multicultural science and math connections:  Middle school projects and activities (pp. 140-143).  Portland, ME:  J. Weston Walch.

 

National Council of Teachers of Mathematics.  (2000).  Principles and  standards for school mathematics.  Reston, VA: Author.

 

Shirley, L. (1988).  Counting in Nigerian languages.  Paper presented at the Sixth International Congress of Mathematical Education, Budapest, Hungary.  (Available from LShirley@towson.edu)

 

Smith, S. (1996).  Agnesi to Zeno:  Over 100 vignettes from the history of math.  Berkeley, CA:  Key Curriculum Press.

 

Williams, S. (1999a).  Benjamin Banneker 1731-1806 – Mathematicians of the African diaspora  [On-line].  Available: http://www.math.buffalo.edu/mad/special/banneker-benjamin.html#bannekerletter

 

Williams, S.  (1999b).  Mathematicians of the African diaspora  [On-line].  Available: http://www.math.buffalo.edu/mad/index.html

 

Williams, S.  (1999c).  Myths, lies, and truths about mathematicians of the African Diaspora  [On-line].  Available:  http://www.math.buffalo.edu/mad/myths_lies.html

 

Williams, S. (1999d).  Thomas Fuller, African slave and mathematician [On-line].  Available: http://www.math.buffalo.edu/mad/special/fuller_thomas_1710-1790.html

 

Zaslavsky, C. (1973).  Africa counts:  number and pattern in African culture. Boston:  Prindle, Weber & Schmidt.


Activity Sheet 1: Yoruba Numeration and Calculation Algorithms

 

A.  Here is a list of some counting words in the Yoruba language of Nigeria (sometimes these are spelled differently).  Study these words and look for patterns.  The “teens” are especially interesting.  Can you find a pattern in the numbers 11 through 14?  How are the numbers 15 through 19 represented?  Twenty is a special number in the Yoruba language.  Compare the numbers that are multiples of twenty (20, 40, 60, …)  with the first ten numbers.   How do the multiples of ten that are not multiples of twenty (30, 50, 70, …) fit into the pattern?


  1        okan

  2        eeji

  3        eeta

  4        eerin

  5        aarun

  6        eefa

  7        eeja

  8        eejo

  9        eesan

 10       eewa

 11       okanla

 12       eejila

 13       eetala

 14       eerinla

 15       aarundinlogun

 16       eerindinlogun

 17       eetadinlogun

 18       eejidinlogun

 19       okandinlogun

 20       oogun

 21       ookan le logun

 22       eeji le logun

 

 28       eeji din logbon

 29       okan din logbon

 30       ogbon

 31       okan le logbon

 38       eeji din logoji

 40       ogoji

 50       aadota

 60       ogota

 70       aadorin

 80       ogorin

 90       aadorun

100       ogorun

110       aadofa

120       ogofa

130       aadoja

140       ogoja

150       aadojo

160       ogojo

170       aadosan

180       ogosan

190       aadowa

200       igba (note: this special number does not fit into the pattern) 


 

How would you write the following in Yoruba words?

a. 33

b. 37

c. 46

d. 54

e. 85

f. 107

g. 136

h. 164

i. 192

j. 199

 

k.  Yoruba people count in groups of twenty.  An old English “score” meaning “twenty” shows that English speakers have also used groups of twenty.  What famous speech that was delivered in 1863 used the word “score” in a number sense in its opening words?

l.  Convert the opening words of the speech into a number.

m.  What year does the speech refer to if you know that it was delivered in 1863? 

 

B.  Sometimes you can speed up mental calculations by using the distributive law.  For example, 12 x 35 can be thought of as (10 + 2) x 35 = 10 x 35 + 2 x 35 = 350 + 70 = 420.

Complete these multiplications mentally as you practice using the distributive law.

a. 11 x 43          b. 32 x 51         c. 105 x 24

 

Sometimes we can use the distributive law with subtraction.  For example,

19 x 46  =  (20 – 1) x 46  =  (20 x 46) – (1 x 46)  =  920 – 46  =  874.

Try to compute these multiplications mentally with the distributive law and subtraction.

d. 39 x 45          e. 98 x 121        f. 108 x 269

 

Perhaps Thomas Fuller used the distributive law in combination with his own counting system in order to complete quick mental calculations.


Activity Sheet 2: Thomas Fuller and Calculation Time

 

Thomas Fuller was born in Africa in 1710.  At the age of 14, he was sold into slavery and he was taken to Virginia.  While he never learned how to read or write, he had an amazing ability to perform calculations in his head.  When Fuller was 78 years old, he was interviewed.  White abolitionists asked him to perform a number of calculations.  One of the questions was to compute the number of seconds a man who is 70 years, 17 days, and 12 hours old has lived.  Astoundingly, he answered 2,210,500,800 (the correct answer) in only a minute and a half. According to a newspaper account:

One of the gentlemen, who employed himself with his pen in making these calculations, told him [Fuller] he was wrong, and that the sum was not so great as he had said—upon which the old man hastily replied, ‘top, massa, you forget de leap year.’  On adding the seconds of the leap years to the others, the amount of the whole in both their sums agreed exactly.”

In addition, when one of the men remarked that it was a shame that Fuller had never had a formal education, Fuller replied, “No, massa, it is best I got no learning; for many learned men be great fools.”  One should remember that Fuller’s style of speech in this account was typical of white accounts of slaves during this time.  In addition, it would have been unusual for an African American slave to correct or contradict a white man, and the fact that Fuller did so is worth contemplation.

 

Problem 1:  Fuller was also asked to calculate the number of seconds in a year and a half, and he answered the problem correctly in approximately 2 minutes.  Using your calculator, find the answer and time yourself to see how long it takes.

Problem 2:  Use the following steps to find the number of days in your lifetime.

Part A:  In order to calculate how many days have you lived, first try to answer this just in your head – no calculator, computer or paper allowed!  Time yourself to see how long it took you to do this in your head and write down the answer and your time here.

Part B:  Now time yourself on paper.  Show your work and write down how long it took.

Part C:  Now pair up with a partner.  Using your calculator, figure out how many days your partner has lived, and time yourself again.

Part D:  Compare your answers with your partner’s answers on Parts B and C.  If they do not match, then go back over them to see which is correct.

Problem 3:  Conduct some research on the Web in order to find out about the history of early calculation machines and computers such as ENIAC.  Could the first calculation machines and computers have beaten Fuller’s calculation times?

Problem 4:  Could modern calculators and computers beat Fuller’s calculation times?

Problem 5:  The calculation speed of computers improves each year.  Find out the speed of a computer at home, at school, or in the library. 

Problem 6: Do you think that there is a limit to how fast the human mind can calculate?  Do you think that there is a limit to how fast computers will be able to calculate in the future?


Select Teacher Solutions

 

Activity Sheet 1

A. For the numbers 11-14, “-la” is added to the numbers 1-4, so twelve is like “two-la”.

For the numbers 15-19, “dinlogun” means “less than ‘ogun’” or “less than twenty” so, for example 16 is called “4 less than 20.”

The names of multiples of twenty are “ogo” and the multiple number.  For example, sixty (3 x 20) is “ogo + ta” from “ogun” (twenty) and “eeta” (three).

The other multiples of ten start with “aad” and then make reference to the next multiple of ten.  For example, fifty (3x20 – 10) is “aadota” where the “-ota” refers to 60 (3 x 20), the next multiple of twenty.  Again, the “ta” means “three.”

    Some of the following answers might vary slightly.

     a. eeta le logbon

     b. eeta din logoji

     c. eerin din laadota

     d. eerin le aadota

     e. aarun din laadorun

     f. eeta din laadofa

     g. eerin din logoja

     h. eerin le ogojo

     i. eeji le aadowa

     j. okan din ligba

     k. Gettysburg Address (by Abraham Lincoln) opens with “Four score and seven years ago…”

     l. (4 x 20) + 7 = 87 years ago

     m. 1863 – 87 = 1776 (the date of the Declaration of Independence)

 

B. These should be done mentally if possible, or with only a few written notes.

     a. 11 x 43 = (10 + 1) x 43 = (10 x 43) + (1 x 43) = 430 + 43 = 473

     b. 32 x 51 = (30 + 2) x 51 = (30 x 51) + (2 x 51) = 1530 + 102 = 1632

            (note: students might break up 51 instead of 32; that’s fine!)

     c. 105 x 24 = (100+5) x 24 = (100 x 24) + (5 x 24) = 2400 + 120 = 2520

     d. 39 x 45 = (40 – 1) x 45 = (40 x 45) – (1 x 45) = 1800 – 45 = 1755

     e. 98 x 121 = (100 – 2) x 121 = (100 x 121) – (2 x 121) =  12100 – 242 = 11858

     f. 108 x 269 = (110 – 2) x 269 = (110 x 269) – (2 x 269) = 29590 – 538 = 29052

         or  (100 + 8) x 269 = (100 x 269) + (8 x 269) = 26900 + 2152 = 29052

            (These are both pretty hard to do completely mentally!)

 

Activity Sheet 2

Problem 1:  47,304,000

Problem 2:  This will depend on the student.  Don’t forget leap years!

Problem 3:  Early calculation machines did not even have enough digits to be able to perform this kind of calculation.  ENIAC could perform 300 multiplication problems per second, which is much faster than Fuller.  Yet, Fuller was able to compute answers larger than ten decimal places, whereas ENIAC could not.  In addition, the creation of punch cards for ENIAC would have added to the time required.

Problem 4:  While modern calculators and computers could complete the computations much faster than Fuller, the human operating the calculator or computer has to type the problem in, which will take some time.

Problem 5:  The CPU (Central Processing Unit) affects calculation speed. 

Problem 6:  This is an open-ended question, asking for the student’s opinion.  One limiting factor might be the speed of light. The new field of quantum computing may eventually allow computations beyond even that limit.