Working Title: The Ethnomathematics of Vietnamese
Algorithms
Daniel C. Orey & Kieu
T. Nguyen
This paper represents the second in an ongoing
series of research findings related to the Algorithm
Collection Project at
Historical Background
Many
foreign countries have influenced the Vietnamese educational system, with the
Chinese and the French making the greatest impact of all in the shaping of the
current academic structure in
The
Vietnamese educational system of the French Indochinese colonial era ended when
Age 
Year/Grade 
22 + 
Colleges, Universities 
21 + 

20 + 

19 + 

18 + 

17 + 
12 11
10 
16 + 

15 + 

14 + 
9 8
7 6 
13 + 

12 + 

11 + 

10 + 
5 4 3
Primary School 2 1 
9 + 

8 + 

7 + 

6 + 

5 + 
‘Young Shoot’ education 
Primary school: Children
from 6 years of age are admitted to this level and their age is calculated
according to the year of birth. Grade 1
is the first grade of primary education, which includes grades 1 – 5.
Junior secondary schools: Children
from the age of 11, consists of three years (grades 6 – 9).
Senior secondary schools: Children
from the age of 15 and consists of three grades (grades 1012).
Higher education: (college, universities and postgraduate
education): Course of study is 3 years
for college, 4 to 6 years for universities. To receive a MA degree, an
individual needs to have graduated from a university; the study is 2
years. For a Doctoral degree, the study
is 3 – 4 years or more.
The universalization
of primary education in
(1) state schools organized by the state (98 % of the pupils
attend these schools) and
(2) private schools
which are organized and managed by individuals (Do, 2003). In both systems, students are given
instruction under a strict environment.
Both Do
(2003) and Phung (2003) have described that in
Vietnam, the majority of teachers use a very rigid or standardized teaching
style as compared to that common in the United States. Concepts are taught first, followed by a
great deal of practice with less emphasis on context. The teacher spends 99% of class time using
highly structured lessons used to complete daily objectives. Often learning is accomplished through the
use of homework.
In the
primary classroom, students are seated in rows.
Typically the mathematics lesson time consists of 90% of whole class
teaching when the teacher places emphasis on the explanation of illustrated
methods to the whole class; 8% is devoted to individual work and 2% to group
work (Do, 2003). During each math
lesson, pupils take turns at working at the board in front of the entire
class. Students’ mistakes are usually
dealt with individually. Basic
operations like addition and multiplication facts are learned by heart. Students are required to solve basic
algorithms through the use of mental math strategies which we will attempt to
explain later in this discussion.
Assessment
Students
in
Vietnamese
immigrants tell us that it appears that the content of the Vietnamese
Educational system is more rigorous than that found in the
The
factors that have been outlined above contribute toward success in Vietnamese children
in math (i.e. mental calculation, hard study, and the overwhelming value
placedon education). It appears from
our research, that the Vietnamese education system does not make it possible
for every student to be successful, only the “smart and quick” kids are able to
take advantage or overcoming the system.
As for the poor, or those who need some form of extra assistance, these
students are left behind and drop out of school.
Calculation and Algorithms
The cultural aspects of
schooling and language as described above indicate to us that the process of
learning mathematics in
In the
Let us look at some of
the examples of the process by which Vietnamese students solve the basic
algorithm of addition, subtraction, multiplication, and division compared to
the method commonly found in the
ADDITION:
Vietnamese’s
Method United
States Method
1 1
2 8 7 2 8 7
+ 4 7 3 + 4 7 3
7 6 0 7 6 0
The process of solving a
multidigit addition problem in
SUBTRACTION:
Vietnamese’s Method
3 11
4 2 7
4 2 7
 1 8 9 1
8 9
2 3 8
2 3 8
In using the Vietnamese
method, one starts in the one’s column and begins by subtracting 9 from 7. But 7 is not big enough, so one borrows ten
from the 2 on the tens column which will then give 17. Now “mentally take away” 9 from 17 to obtain
8. Then add 1 back to the 8 on the tens
column to get 9 and mentally subtract 9 from 2.
Again 2 is too small to subtract 9 so borrow another ten from the
hundreds column. You now have 12 minus 9
to get 3. Now, move to
the hundreds column and return 1 to the 1 in the hundreds column to get 2. Then mentally subtract 2 from 4 and get
2. Your final answer is 238. The differences between the Vietnamese method
and the method as traditionally taught in the United States of doing
subtraction seems to be that Vietnamese calculation borrows from the ten from
the top number but returns 1 back to the bottom number. In the
MULTIPLICATION
Vietnamese’s Method
1 2
1 2 3 1 2 3
x 7 x 7
8 6
1 8 6 1
The
Vietnamese process of multiplying is similar to that of addition and
subtraction discussed above as students are still required to do mental
math. Begin by multiplying 3 and 7 to
get 21 then put the 1 under the line, but do not write the 2 over the 2 on the
tens column (don’t carry the 2 over).
Next, multiply 2 and 7 in your head and add 2 that you mentally
carriedover from the previous calculation to get 16. Write the number 6 in the tens column
underneath the line and disregard writing the 1 that needs to be carried
over. Then you multiply 7 and 1 then add
1 in your head to get 8. In this method
the only number written is the answer and you don’t write the numbers that are
carried over to the next column.
Finally, to make sure the answer is correct, Vietnamese mathematics
involves an interesting checking algorithm for correctness.
1 2 3 (a)
x 7 (b) (a) 6
8
6 1 (d)
(d)
6 (c) 6
(b) 7
1. Start
by drawing an X.
2. Locate (a) in the multiplication problems,
which is the top number and add all three digits, which are 1, 2, and 3 to get
6. Write that number in the space that
is labeled (a) in the X.
3. Then add up all the digits indicated by the
(b) which is only one digit the number 7. Write that number on the bottom of
the X labeled (b).
4. Now multiple the numbers in the space of (a)
and (b) in the X, which is 6 and 7 to get 42.
Then add the product of (a) and (b) which is 4 + 2 = 6. Write the number 6 in the X labeled (c).
5. Finally, you want to add all the digits of
the answer labeled (d) from the problem above, which are 8, 6, 1 and you, get
15. Then add the 2 digits of your sum
that is 1 +5 =6. Write the 6 in the
space labeled (d) in X.
6. If the numbers from (c) and (d) are the same
then the answer you got from the original problem is correct. In this case (c) = 6; and (d) = 6, the two
answers match.
DIVISION:
Vietnamese
method
24
2 1 2
04
12 2 2 4
0  2
0 4
 4
0
In
Vietnamese, the division bar is drawn differently from that used in the
Checking your answer
(c) 24
2 (a) (a) 2
12 (b)
(c ) 6 (d) 6
(b)
3
1. As in multiplication, draw a big X.
2. Write
the number indicated by (a) from the division problem to the space labeled (a)
in the X which is 2.
3. Add
up the two digits of the quotient, which is 12 (1 + 2) to get 3. Write this number in the space labeled (b) in
the X.
4. Add
the two digits 24 (dividend) together which is labeled (c), so you get 2 + 4 =
6. Write 6 in the space labeled (c) in
the X.
5. You
multiply the number from (a) and (b) which is 2 and 3 to get 6. Place the
number 6 in the space labeled (d) in the X.
6. If the number from (c)
and (d) are the same then the answer is correct.
Conclusion
Ms.
Nguyen, a VietnameseAmerican, and coauthor grew up in the
What we
found is that now there is an increased value on education both here in the
As an
American of Vietnamese background educated in the
Through
a study of the basic algorithms and procedures as used in
The
Algorithm Collection Project[i]
by using an ethnomathematicsbased pedagogy is interested in studying the
observable linkages between how particular algorithms are used, what and how
many languages are spoken, and the internalization for the mathematics used by
an individual.
Bibliography
Do,
Dat. (n.d). Mathematics
teaching and learning in Vietnam. Retrieved on March 14,
2003 from http://www.ex.ac.uk/cimt/ijmtl/ddvietmt.pdf
Sloper, David & Can, Le T. (1995). Higher education in Vietnam: Change and response.
Q. Phong,
personal communication, March 14, 2003.
The authors greatly appreciate the
feed back from Charlie Carroll at the University of California, Berkeley. Quoc Phung, Pamela Milchrist and Vicki PearsonRounds at California State
University, Sacramento, and Ubiratan D’Ambrosio in São Paulo, Brazil