Better For Problem Solving?
Why Are Visualizations & Drawing As Important As Numbers?
Plus Bonus Video Below
If you missed Bill's fascinating journey, from Japan to Germany, discovering and pursuing Singapore Math, see Part 1 - "Why I Became Interested in Singapore Math". For Part 2 - "Can Problem Solving Unravel Our Fear of Math?" . Part 4 "Bringing Singapore Math to your School & Tips for Teachers" . ---C.J. Westerberg
What comes to mind when you think about math word problems? If you are among
the lucky few, they may not cause too much anxiety. But for many if not most people,
they evoke memories of frustration, failure and dislike of mathematics.
This is strange considering the fact that problem solving has been a major focus of
math education in the U.S. since at least 1989 with the publication of the Principles
and Standards for School Mathematics by the National Council of Teachers of
Mathematics. Since then, virtually every state in the country has developed math
standards that include statements like this one from New York: "mathematics
instruction must include the teaching of many strategies to empower all students to
become successful problem solvers." U.S. math textbooks even devote entire
sections to learning strategies for solving word problems like "guess and check,"
"work backwards," "make a table," and "act it out." But every day American students
are confused, discouraged, and unable to solve them.
Why do word problems cause so much difficulty? Many people struggle with word
problems because words themselves are abstract. In 1977, Australian educator
Anne Newman discussed five steps that students need to work through in order to
solve a word problem successfully--
(1) reading the problem
(2) comprehending what was read
(3) transforming the words into a mathematical strategy
(4) applying a mathematical procedure and
(5) writing the answer.
Her research showed that over 50% of errors that children make occur in the first three steps--
before they even begin to solve the problem!
In order to help children understand word problems, teachers often focus on key
words such as "more" and "times." This strategy is useful but limited because key
words don't help students understand the problem situation (i.e. what is happening
in the problem). Key words can also be misleading because the same word may
mean different things in different situations. Consider the following two examples:
There are 7 boys and 21 girls in a class. How many more girls than boys are
There are 21 girls in a class. There are 3 times as many girls as boys. How
many boys are in the class?
In the first problem, if students focus on the word "more" they may add when they actually need to subtract. In the second problem, if students focus on the word "times," they may multiply when they actually need to divide.
Instead of relying on ambiguous key words, Singapore math textbooks help students
to visualize problem situations by turning abstract words into easy to understand
pictorial models. By constructing a model we can understand the problem situation
Let's think about the above problems again using a "bar model," the most common
(but not the only) pictorial model used in Singapore math textbooks.
To illustrate and understand the problem, let's draw two bars. There are more girls than boys, so we'll draw a shorter bar to represent the boys and a longer bar toThere are 7 boys and 21 girls in a class. How many more girls than boys are
represent the girls. The question mark indicates what we are trying to find out--the
difference between the number of boys from the number of girls.
The diagram shows clearly that the answer is less than 21 (so adding the two
need to subtract 7 from 21 to find the answer.
Now let's use a bar model to think about the second problem.
Again we'll draw two bars to illustrate the situation, one for the boys and another one exactlyThere are 21 girls in a class. There are 3 times as many girls as boys. How
many boys are in the class?
three times as long for the girls (since there are three times as many girls). We are trying to find the number of boys, so we'll place a question mark
above the part representing the boys.
amounts won't work). By looking at the bar model we can see that 3 units (parts) =
21. Since we want to find 1 unit (the number of boys) we should divide 21 by 3 to
get the answer.
According to Singapore's Handbook for Mathematics Teachers in Primary Schools,
this model drawing approach is helpful for several reasons:
(1) It "helps pupils visualize situations"
(2) It "creates concrete pictures from abstract situations."
(3) It "satisfies the pupils' learning through seeing and doing."
(4) It "transforms words into recognizable pictures for young minds."
Two basic models are used to represent problem‐solving situations in Singapore
math--the part‐whole model and the comparison model. The particular model that
is used depends on the problem situation. The above examples use a "comparison
model" because they involve a comparison of two quantities--the number of boys and the number of girls. When the word problem involves a whole and its parts, however, we use a part‐whole model as in the following example.
Frank has 250 baseball and football cards altogether. He has 110 baseball
card. How many football cards does he have?
In this problem we know the total number of cards (the whole) and the number of
baseball cards (one part). We are trying to find the missing part (the football cards).
By looking at the diagram we can see that 110 + ? = 250, so to find the answer we
should subtract 110 from 250.
The usefulness of bar models becomes more apparent as students tackle more
difficult mathematical concepts. Below is an example of how a bar model can be
used to solve a challenging problem from the fifth grade Primary Mathematics
textbook involving fractions, a topic many students struggle with.
Meihua spent 1/3 of her money on a book. She spent 3/4 of the remainder on a pen. If the pen cost $6 more than the book, how much money did she spend altogether?
Let's draw a bar to represent all of Meihua's money. We know that she spent 1/3 of her money on a book, so we'll divide it into thirds. One part (1/3) represents the money she spent on the book. The other two parts represent the remainder of her money.Solving a problem of this level of difficulty would normally require a considerable knowledge of algebra. It can be solved easily by 5th grade
students, however, with a bar model.
The problem says that she spent 3/4 of the remainder on a pen. We know that 3/4 is 3 out of 4 equal parts, so let's cut each of the yellow sections in half to make 4 equal parts (the yellow section). Three of these parts represent the money she spent on
If we make all the parts the same size the problem will be easier to solve because if we can find the value of one unit, all the others will be the same. So let's divide the part representing the book into two equal parts also.
We now have a bar divided into six equal‐sized units (parts), three representing the
pen and two representing the book. We want to find the amount she spent on both
items, which is 5 units.
From the diagram we can see that the difference between the pen and the book is one unit (the 3 pen units - the 2 book units). We know from the problem that this amount is equal to $6 since the pen costs $6 more than the book.
Remember that all the units are the same size and thus have the same value. If we
can find the value of one unit, then we will know the value of all of them. So we can
use the following method.
Notice that this problem can be solved with simple multiplication; no algebra or1 unit = $6
5 units = 5 x $6 = $30
Meihua spent $30 altogether.
complex procedures involving fractions required.
It takes time to develop model‐drawing skills in students, but it is well worth the
effort. As children become increasingly proficient at constructing models they gain
confidence in their problem‐solving abilities. Proficiency in the use of model
drawing helps students to solve increasingly complex problems as they progress
through elementary and middle school mathematics and eventually make the
transition from arithmetic to algebra with greater ease.
Some U.S. mathematics textbooks also incorporate model drawing, but many
different models are typically introduced in a haphazard and unconnected fashion
with inadequate development. The result is that the type of model drawing used in
Singapore is essentially non‐existent in the U.S. and few American students or even
teachers ever learn this powerful problem‐solving technique.
Singapore math textbooks, on the other hand, present and develop a few
consistently used models in a clear and systematic way. The result is that students
become adept at converting abstract word problems into concrete pictures that are
easily translated into mathematical procedures. A recent study reported that
Singaporean students are exposed to higher‐level, multi‐step word problems than
are U.S. students, and proficiency in solving these complex problems is a key factor
in why they have fared so well on international mathematics assessments.
If children are to learn how to solve word problems effectively, teachers need better
ways to help them understand them. Pictorial models are an important intermediate
step between the difficult transition of reading a word problem to determining the
operations and steps necessary to solve it. Singapore's model drawing approach
helps children to get past the words by visualizing and illustrating word problems
with simple diagrams. And as children become better and more confident problem
solvers, they become more interested in mathematics.
That's all for now. If you would like more information on solving problems with bar
models, I have attached a power point presentation that will help you hone your
model drawing skills. Next time I'll share some tips for schools that are interested in
using Singapore math.
Math Helping Teacher
Scarsdale Public Schools
Orig. Published April 2010 The Daily Riff
Part 1: Singapore Math: Can It Solve Our Country's Math-phobia?
Part 2: Singapore Math: Can Solving Problems Unravel Our Fear Of Math?
Part 3: "Singapore Math: Is this the Most Visual Math?" The famous bar models
Part 4: Bringing Singapore Math to Your School + Tips For Teachers
Also check out Bill Jackson's recent travel Journal: "Singapore: Five Surprises In Education"
Related: Bill Jackson's Travel Journal: "What American Teachers Can Learn From Japan"
Check out Bill Jackson's new Travel Journal to Japan for "Lesson Study" HERE.
Check out 2min. video from a Singapore math school in Phoenix - Channel 12 - below: