Math

Singapore Math - The Most Visual Math?

CJ Westerberg, March 1, 2013 8:57 PM

math.singapore.3.abstract.jpg

  Better For Problem Solving?
Why Are Visualizations & Drawing As Important As Numbers? 

 
Plus Bonus Video Below

Editor's Note: Math-phobia is one of the more challenging issues for our students and for this country's educational make-up and there is much to learn from other leading countries such as Singapore.
                  
This is the third installment of the 4-part series "Singapore Math Demystified!" from The Daily Riff,
guest posted by Bill Jackson, Math Helping Teacher, Scarsdale, NY Public Schools, one of the highest performing districts in the country, due to the remarkable interest expressed by parents and schools.  We are delighted that he is sharing his wealth of knowledge on this subject with us!

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


The Singapore Math Model‐Drawing Approach

By Bill Jackson 
 
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? 
  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 
more clearly.
  
 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.
 
  There are 7 boys and 21 girls in a class. How many more girls than boys are 
   there?
 
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 to 
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.

Picture 10.pngThe diagram shows clearly that the answer is less than 21 (so adding the two 
mounts definitely won't work). By looking at the model it is easy to see that we 
need to subtract 7 from 21 to find the answer.
 
Now let's use a bar model to think about the second problem.
 
  There are 21 girls in a class. There are 3 times as many girls as boys. How 
   many boys are in the class?
 
 Again we'll draw two bars to illustrate the situation, one for the boys and another one exactly
 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.

Picture 3.pngThe diagram shows that the answer has to be less than 21 (so multiplying the 
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?

Picture 4.png
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?
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.

 
 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.

Picture 11.png 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 
the pen.

Picture 6.png 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.


Picture 8.pngWe 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. 

Picture 9.png
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.
 
   1 unit = $6 
   5 units = 5 x $6 = $30 
   Meihua spent $30 altogether.  
Notice that this problem can be solved with simple multiplication; no algebra or 
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.
 
Until then,  
Bill Jackson 
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:




 

  • blo wee

    I can see and also I think your the type of person who want's to learn and the person that really works hard for the knowledge :) ,, Also I heard a lot of good programs about singapore math ..

    http://merlenemath.com/

  • Ng

    I am a educational counsultant / maths tutor from Singapore.

    Singapore maths has got a lot of good features. The curriculum

    planners / developers at Singapore's Ministry of Education

    systematically applied Bruner and other sound educational

    theories to design the curriculum. It is systematic, it (tries

    to) deal with concepts, and (tries to) introduce problem

    solving techniques. Our teachers set very high standards for

    the students -- no excuses for having been born with a "low

    IQ".

    However, be careful not to over-worship the curriculum. The

    curriculum is as good as its implementation. My private

    tutoring experience seems to suggest that Singapore's school

    teachers don't really teach that well, especially from

    secondary school onwards. I feel that we work very hard, but

    not very smart. Our students spend many hours on homework

    (without necessarily understanding what they are doing) and

    home tuition is very popular in Singapore. But in Finland, the

    students do not have home tuition and they only need to spend a

    few hours in school each day AND THEY "GET IT" QUICKLY. That's

    because their teachers are very good!

    So the quality of the teacher matters. actually, I would like

    to throw down the following challenge: give me the "worst"

    student and the "worst" curriculum. see if i cannot teach

    him/her to learn mathematics well. anyone game for this

    experiment?

    Anyway, do you want to see some real Singapore maths (not just

    at elementary school level)? Please visit my blog

    http://genuinesingaporemaths.blogspot.com

    though very good, Singapore's Maths (and indeed the Maths

    Education all over the world) itself needs fixing. here are my

    ideas

    http://genuinesingaporemaths.blogspot.com/2012_01_01_archive.ht

    ml

    [ in Singapore, it's just called "maths". ]

  • Lisa T.

    Mr. Jackson, I have heard SO much about Singapore Math. In fact, I'm doing a presentation tomorrow----trying to sell the idea to our school board. We currently use Saxon. Could I use your powerpoint as part of my presentation. I think it would truly benefit our cause!!

  • Thanks Jeff - would like to hear more about your students who did take Singapore Math vs. those who didn't . . . .

  • jeff Dennis

    We are long overdue for Singapore Math-I wish I could collect all my former elementary school students and introduce them to the "concrete", the pictorial bar model, and then teach them the algorithm. Bill Jackson is absolutely correct about Singapore Math. I wish our country's schools would take the time to understand and then teach using math the Singapore Math approach. I am fortunate-my school adopted Singapore Math across all K-5 grade levels two years ago. It took rigorous work by the teaching staff-but the staff took it on and the kids are really learning with one of the best methods in the world.

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