The Mathematics Educator
2004, Vol. 14, No. 1, 42–46
Sybilla Beckman 42
Solving Algebra and Other Story Problems with Simple
Diagrams: a Method Demonstrated in
Grade 4–6 Texts Used in Singapore
Sybilla Beckmann
Out of the 38 nations studied in the 1999 Trends in International Mathematics and Science Study (TIMSS),
children in Singapore scored highest in mathematics (National Center for Education Statistics, NCES, 2003).
Why do Singapore’s children do so well in mathematics? The reasons are undoubtedly complex and involve
social aspects. However, the mathematics texts used in Singapore present some interesting, accessible problem-
solving methods, which help children solve problems in ways that are sensible and intuitive. Could the texts
used in Singapore be a significant factor in children’s mathematics achievement? There are some reasons to
believe so. In this article, I give reasons for studying the way mathematics is presented in the elementary
mathematics texts used in Singapore; show some of the mathematics problems presented in these texts and the
simple diagrams that accompany these problems as sense-making aids; and present data from TIMSS indicating
that children in Singapore are proficient problem solvers who far outperform U.S. children in problem-solving.
Why Study the Methods of Singapore’s
Mathematics Texts?
What is special about the elementary mathematics
texts used in Singapore? These texts look very
different from major elementary school mathematics
texts used in the U.S. The presentation of mathematics
in Singapore’s elementary texts is direct and brief.
Words are used sparingly, but even so, problems
sometimes have complex sentence structures. The page
layout is clean and uncluttered. Perhaps the most
striking feature is the heavy use of pictures and
diagrams to present material succinctly—although
pictures are never used for embellishment. Simple
pictures and diagrams accompany many problems, and
the same types of pictures and diagrams are used
repeatedly, as supports for different types of problems,
and across grade levels. These simple pictures and
diagrams are not mere procedural aids designed to help
children produce speedy solutions without
understanding. Rather, the pictures and diagrams
appear to be designed to help children make sense of
problems and to use solution strategies that can be
justified on solid conceptual grounds. Because of this
pictorial, sense-making approach, the elementary texts
used in Singapore can include problems that are quite
complex and advanced. Children can reasonably be
expected to solve these problems given the problem-
solving and sense-making tools they have been
exposed to.
Thus the strong performance of Singapore’s
children in mathematics may be due in part to the way
mathematics is presented in their textbooks, including
the way simple pictures and diagrams are used to
communicate mathematical ideas and to provide sense-
making aids for solving problems. If so, then teachers,
mathematics educators, and instructional designers in
the U.S. will benefit from studying the presentation of
mathematics in Singapore’s textbooks, so that they can
help children in the U.S. improve their understanding
of mathematics and their ability to solve problems.
Using Strip Diagrams to Solve Story
Problems
One of the most interesting aspects of the
elementary school mathematics texts and workbooks
used in Singapore (Curriculum Planning and
Development Division, Ministry of Education,
Singapore, 1999, hereafter referred to as Primary
Mathematics and Primary Mathematics Workbook) is
the repeated use of a few simple types of diagrams to
aid in solving problems. Starting in volume 3A, which
is used in the first half of 3rd grade, simple “strip
diagrams” accompany a variety of story problems.
Consider the following 3rd grade subtraction story
problem:
Sybilla Beckmann is a mathematician at the University of
Georgia who has a strong interest in education. She has
developed three mathematics content courses for prospective
elementary teachers and has written a textbook, Mathematics for
Elementary Teachers, published by Addison-Wesley, for use in
such courses. In the 2004/2005 academic year, she will teach a
class of 6th grade mathematics daily at a local public middle
school.
Sybilla Beckman 43
Mary made 686 biscuits. She sold some of them. If
298 were left over, how many biscuits did she sell?
(Primary Mathematics volume 3A, page 20,
problem 4)
The problem is accompanied by a strip diagram like
the one shown in Figure 1.
Figure 1: How Many Biscuits Were Sold?
On the next page in volume 3A is the following
problem:
Meilin saved $184. She saved $63 more than Betty.
How much did Betty save? (Primary Mathematics
volume 3A, page 21, problem 7)
This problem is accompanied by a strip diagram like
the one in Figure 2.
Figure 2: How Much Did Betty Save?
These two problems are examples of some of the
more difficult types of subtraction story problems for
children. The first problem is difficult because we must
take an unknown number of biscuits away from the
initial number of biscuits. This problem is of the type
change-take-from, unknown change (see Fuson, 2003,
for a discussion of the classification of addition and
subtraction story problems). The second problem is
difficult because it includes the phrase “$63 more
than,” which may prompt children to add $63 rather
than subtract it. This problem is of type compare,
inconsistent (see Fuson, 2003). The term inconsistent is
used because the phrase “more than” is inconsistent
with the required subtraction. Other linguistically
difficult problems, including those that involve a
multiplicative comparison with a phrase such as “N
times as many as”, are common in Primary
Mathematics and are often supported with a strip
diagram. Consider the following 3rd grade problem,
which is supported with a diagram like the one in
Figure 3:
A farmer has 7 ducks. He has 5 times as many
chickens as ducks….How many more chickens
than ducks does he have? (Primary Mathematics
volume 3A, page 46, problem 4)
(Note: The first part of the problem asks how many
chickens there are in all, hence the question mark about
all the chickens in Figure 3 below.)
Figure 3: How Many More Chickens Than Ducks?
Although the strip diagrams will not always help
children carry out the required calculations (for
example, we don’t see how to carry out the subtraction
$184 – $63 from Figure 2), they are clearly designed to
help children decide which operations to use. Instead
of relying on superficial and unreliable clues like key
words, the simple visual diagram can help children
understand why the appropriate operations make sense.
The diagram prompts children to choose the
appropriate operations on solid conceptual grounds.
From volume 3A onward, strip diagrams regularly
accompany some of the addition, subtraction,
multiplication, division, fraction, and decimal story
problems. Other problems that could be solved with the
aid of a strip diagram do not have an accompanying
diagram and do not mention drawing a diagram.
Fraction problems, such as the following 4th grade
problem, are naturally modeled with strip diagrams
such as the accompanying diagram in Figure 4:
David spent 2/5 of his money on a storybook. The
storybook cost $20. How much money did he have
at first? (Primary Mathematics volume 4A, page
62, problem 11)
Without a diagram, the problem becomes much
more difficult to solve. We could formulate it with the
equation (2/5)x = 20 where x stands for David’s
original amount of money, which we can solve by
dividing 20 by 2/5. Notice that the diagram can help us
see why we should divide fractions by multiplying by
the reciprocal of the divisor. When we solve the
problem with the aid of the diagram, we first divide
$20 by 2, and then we multiply the result by 5. In other
words, we multiply $20 by 5/2, the reciprocal of 2/5.
44 Solving Problems with Simple Diagrams
Figure 4: How Much Money Did David Have?
The problems presented previously are arithmetic
problems, even though we could also formulate and
solve these problems algebraically with equations. But
starting with volume 4A, which is used in the first half
of 4th grade, algebra story problems begin to appear.
Consider the following problems:
1. 300 children are divided into two groups. There
are 50 more children in the first group than in the
second group. How many children are there in the
second group? (Primary Mathematics volume 4A,
page 40, problem 8)
2. The difference between two numbers is 2184. If
the bigger number is 3 times the smaller number,
find the sum of the two numbers. (Primary
Mathematics volume 4A, page 40, problem 9)
3. 3000 exercise books are arranged into 3 piles.
The fist pile has 10 more books than the second
pile. The number of books in the second pile is
twice the number of books in the third pile. How
many books are there in the third pile? (Primary
Mathematics volume 4A, page 41, problem 10)
These problems are readily formulated and solved
algebraically with equations, but since the text has not
introduced equations with variables, the children are
presumably expected to draw diagrams to help them
solve these problems. Notice that from an algebraic
point of view, the second problem is most naturally
formulated with two linear equations in two unknowns,
and yet 4th graders can solve this problem.
The 5th grade Primary Mathematics texts and
workbooks include many algebra story problems which
are to be solved with the aid of strip diagrams. Some
do not have accompanying diagrams, but others do,
and some include a number of prompts, such as a
diagram like the one in Figure 5 which accompanies
the following problem:
Raju and Samy shared $410 between them. Raju
received $100 more than Samy. How much money
did Samy receive? (Primary Mathematics volume
5A, page 23, problem 1)
Figure 5: Raju and Samy Split Some Money
Notice that the manipulations we perform with
strip diagrams usually correspond to the algebraic
manipulations we perform in solving the problem
algebraically. For example, to solve the previous Raju
and Samy problem, we could let S be Samy’s initial
amount of money. Then,
2S + 100 = 410
as we also see in Figure 5. When we solve the problem
algebraically, we subtract 100 from 410 and then
divide the resulting 310 by 2, just as we do when we
solve the problem with the aid of the strip diagram.
Strip diagrams make it possible for children who
have not studied algebra to attempt remarkably
complex problems, such as the following two, which
are accompanied by diagrams like the ones in Figure 6
and Figure 7 respectively:
Encik Hassan gave 2/5 of his money to his wife
and spent 1/2 of the remainder. If he had $300 left,
how much money did he have at first? (Primary
Mathematics volume 5A, page 59, problem 6)
Raju had 3 times as much money as Gopal. After
Raju spent $60 and Gopal spent $10, they each had
an equal amount of money left. How much money
did Raju have at first? (Primary Mathematics
volume 6B, page 67, problem 1)
Sybilla Beckman 45
Figure 6: How Much Money Did Encik Hassan Have at
First?
Figure 7: How Much Did Raju Have at First?
Performance of 8th Graders on TIMSS
In light of the complex problems that children in
Singapore are taught how to solve in elementary
school, the strong performance of Singapore’s 8th
graders on the TIMSS assessment is not surprising.
Among the released TIMSS 8th grade assessment
items in the content domain “Fractions and Number
Sense” classified as “Investigating and Solving
Problems,” Singapore 8th graders scored higher than
U.S. 8th graders on all items. These released items
included the following problems (see NCES, 2003):
Laura had $240. She spent 5/8 of it. How much
money did she have left? (Problem R14, page 29.
Overall percent correct, Singapore: 78%, United
States: 25%).
Penny had a bag of marbles. She gave one-third of
them to Rebecca, and then one-fourth of the
remaining marbles to John. Penny then had 24
marbles left in the bag. How many marbles were in
the bag to start with?
A. 36
B. 48
C. 60
D. 96
(Problem N16, page 19. Overall percent correct,
Singapore: 81%, United States: 41%)
These problems are similar to problems in Primary
Mathematics. The strong performance of Singapore 8th
graders on these problems indicates that the instruction
children receive in solving these kinds of problems is
effective. Similarly, among the released TIMSS 8th
grade assessment items in the content domain
“Algebra” classified as “Investigating and Solving
Problems,” Singapore 8th graders scored higher than
U.S. 8th graders on all items.
But the strong problem-solving abilities of
Singapore’s 8th graders in fractions and number sense
and in algebra does not necessarily result in factual
knowledge in other mathematical domains in which the
children have not had instruction. For example, U.S.
8th graders scored higher than Singapore 8th graders
on the following item in the content domain “Data
Representation, Analysis and Probability” classified as
“Knowing”:
If a fair coin is tossed, the probability that it will
land heads up is 1/2. In four successive tosses, a
fair coin lands heads up each time. What is likely
to happen when the coin is tossed a fifth time?
A. It is more likely to land tails up than heads up.
B. It is more likely to land heads up than tails up.
C. It is equally likely to land heads up or tails up.
D. More information is needed to answer the
question.
(Problem F08, page 74. Overall percent correct,
United States: 62%, Singapore: 48%)
The mathematics texts used in Singapore through
8th grade do not address probability. Thus the
difference in performance in fraction, number sense,
and algebra problem-solving versus knowledge about
probability can reasonably be attributed to effective
instruction.
46 Solving Problems with Simple Diagrams
Conclusion
The mathematics textbooks used in elementary
schools in Singapore show how to represent quantities
with drawings of strips. With the aid of these simple
strip diagrams, children can use straightforward
reasoning to solve many challenging story problems
conceptually. The TIMSS 8th grade assessment shows
that 8th graders in Singapore are effective problem
solvers and are much better problem solvers than U.S.
8th graders. Although cultural factors probably also
affect the strong mathematics performance of children
in Singapore, children in the U.S. could probably
strengthen their problem-solving abilities by learning
Singapore’s methods and by being exposed to more
challenging and linguistically complex story problems
early in their mathematics education.
REFERENCES
Curriculum Planning and Development Division, Ministry of
Education, Singapore (1999, 2000). Primary Mathematics (3rd
ed.) volumes 1A–6B. Singapore: Times Media Private
Limited. Note: additional copyright dates listed on books in
this series are 1981, 1982, 1983, 1984, 1985, 1992, 1993,
1994, 1995, 1996, 1997, thus 8th graders who took the 1999
TIMSS assessment used an edition of these books.
Curriculum Planning and Development Division, Ministry of
Education, Singapore (1999, 2000). Primary Mathematics
Workbook (3rd ed.) volumes 1A–6B. Singapore: Times Media
Private Limited.
Fuson, K. C. (2003). Developing Mathematical Power in Whole
Number Operations. In J. Kilpatrick, W. G. Martin, and D.
Schifter, (Eds.), A Research companion to principles and
standards for school mathematics (pp. 68–94). Reston,VA:
National Council of Teachers of Mathematics.
National Center for Education Statistics (2003). Trends in
international mathematics and science study. Retrieved May
3, 2004, from http://nces.ed.gov/timss/results.asp and from
http://nces.ed.gov/timss/educators.asp
