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**

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!)

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.