## Friday, August 2, 2013

### SINGAPORE MATH: USING BAR MODELS

PARENTS’ GUIDE TO HELPING STUDENTS DRAW BAR MODELS

My mother still laments the transition to “the New Math” in the elementary curriculum between my sister’s experience and my own -- and THAT was 50 years ago!!  It is understandable that parents (and some teachers) are hesitant when something new and radical is implemented in an otherwise comfortably familiar class.

This is the state of affairs in more and more school districts as the educational system searches for ways to improve America’s lack of stellar performance in science, technology, engineering, and math (STEM) programs.  Statistics over the past decade report that very young children in Singapore ranked first in the world in mathematics (Trends in International Mathematics and Science Study (TIMSS)-2003), initiating a wave of interest in the current edition of the newest math experiment, Singapore Math.

I have one suggestion for parents who are feeling unsure of their ability to help their students make the transition to Singapore Math.

DO NOT PANIC!!

Although jumping head first into a new method of “thinking math” can seem an impossible challenge, the “Bar Modeling” method used to visualize word problems in Singapore Math is less complicated than it “appears.”  And “appears” is exactly the right term to use because the Bar Method is intended to be a visual collection of details from a word problem.  It is probably most applicable for students who are visually oriented, Visual Learners, but many schools will and are presenting the method to all students, regardless of their learning style.  In the elementary grades, the philosophy is that the very young are not capable of thinking abstractly and require concrete exemplars.  Supporters of Singapore Math profess that the system is a superior way to lead children to a strong foundation for understanding the basics of math.

For parents new to the subject, the “8 Step Approach” proffered by recognized experts on Singapore Math could be helpful if not somewhat cumbersome.  Boiling the process down to a more streamlined form will provide every parent with the basics to competently help with homework.
1.  Draw a rectangle.
2.  Divide it into boxes.
3.  Color in some boxes.
4.  Divide into smaller boxes.
5.  Count.

While implementing Singapore Math, think of denominators.  They form the basis for dividing up the BARS.  The process can most easily be described through example.

Lily makes macramé bracelets to sell at the local flee market.  Last weekend, she sold 3/5 of her jewelry on Saturday and 1/4 of the rest of her inventory on Sunday.  If she sold 20 more bracelets on Saturday than she did on Sunday, how many bracelets did she have in the first place.

If 20 bracelets are in the 5 dotted BARS, there are 4 bracelets in each BAR.  We're still using fractional division here. There are now a total of 5 times 2, or 10 bars, each holding 4 bracelets, so there were 40 bracelets at the beginning of the weekend. The division of BARS is relatively simplistic in this example.

Here’s the “traditional” format for writing equations to solve this problem.  It relies on direct translations of the instructions in the word problem to “mathese,” a step that is frequently skipped by students in high school courses, but which can add points on an AP test.

X = total bracelets
Saturday Sales = 3/ 5 X
Sunday Sales = 1/ 4(1 X - 3/ 5 X) = 1/4 (2/ 5 X) = 2/ 20 X
Saturday Sales = Sunday Sales + 20

3/ 5 X = 2/ 20 X + 20
12/ 20 X = 2/ 20 X + 20
12/ 20 X - 2/ 20 X = 20
10/ 20 X = 20
X = 20 (20/ 10) = 40 bracelets

But what about divisions that are more complicated.  A few minor changes in the numbers and dividing the remaining bars seems confusing, but there’s a simple technique.

Lily makes macramé bracelets to sell at the local flee market.  Last weekend, she sold 3/5 of her jewelry on Saturday and 1/3 of the rest of her inventory on Sunday.  If she sold 14 more bracelets on Saturday than she did on Sunday, how many bracelets did she have in the first place.

In the first problem, dividing the remaining BARS was a simple task of dividing each into 2.  This time, instead of dividing into simple fourths, we need to divide the two BARS into three and make similar divisions to Saturday’s sales.  How do you divide 2s into 3s? The least complicated way to create subdivisions is to divide EACH BAR into three pieces.  This process is not unlike finding a common denominator.  Notice that the 2 pieces become 6 smaller parts.

Since there are still 7 dotted BARS in Saturday's group prepresenting 14 bracelets, each new subdivision holds 2 bracelets.  There are now 15 subdivisions in all, so Lily had 30 bracelets to sell at the flee market.

The traditional solution looks like this:

X = total bracelets
Saturday Sales = 3/ 5 X
Sunday Sales = 1/ 3 (2/ 5 X) = 2/ 15 X
Saturday Sales = Sunday Sales + 14
3/ 5 X = 2/ 15 X + 14
9/ 15 X = 2/ 15 X + 14
9/ 15 X - 2/ 15 X = 14
7/ 15 X = 14
X = 14 (15/ 7) = 30 bracelets to sell in the first place

The problems demonstrated so far are quite complicated and would appear in Singapore Math around the fifth grade level.  If the student started the process earlier in elementary school, the BAR drawings would be a familiar format that may have started with a very, very simple example of adding fractions like this one...

Lily makes lemonade to sell at a booth in her driveway.  She uses all of her ingredients to make lemonade on Saturday and sells 3/5 of her supply on Saturday and 1/3 of the remainder on Sunday. What fraction of the original lemonade was left at the end of the weekend?

Steps:
1.  Divide whole into 5 parts (to
represent the denominator of 3/ 5)

2.  Count out 3 BARS for Saturday.
3.  Divide the remaining BARS into thirds (for 1/ 3 of remaining juice)
4.  Divide Saturday’s BARS similarly.
5.  Mark out 2/6 of Sunday’s subBARS
6.  Mark out the same amount on Saturday (only because it’s become a habit by now, even though it is NOT NECESSARY).
7.  Count the subBARS which have NOT been marked out.

4/15 of the original lemonade was left over after sales on Sunday.

Saturday Sales = 3/ 5 X
Sunday Sales = 1/ 3 (2/ 5 X) = 2/ 15 X

Saturday Sales + Sunday Sales = 3/ 5 X + 2/ 15 X = 11/ 15 X
Total lemonade - Sales = 1X - 11/ 15 X = 4/ 15 X

#### CON:  I’m really not sure whether this method makes fractions easier for students to learn and eventually manipulate.  My own first exposure to fractions was pre-New Math, so I’ve been experimenting with Singapore Math just like everyone else.  At first, CREATING more fractions seems counterintuitive and may cause consternation in the hearts of parents and students.

PRO:  It may be reassuring to parents to find the Singapore approach very similar to one we’ve used for years to teach early elementary numeracy by using cardboard fraction bars (notice the similarity in terminology) or Legos.  In fact, there are commercial Lego kits to use in teaching Singapore Math.  I am certainly delighted to see the attempt to make fractions into friendly things that can be expressed visually for those who think that way, especially since we use rational expressions to great extent in Algebra, Trigonometry, and Calculus.

CON:  Since experts contend that young students are unable to grasp the theoretic concepts, why aren’t they as concerned with those children’s manual dexterity in creating BARS?  The BAR modeling concept relies on  EQUAL divisions.  Consider this potential student drawing:
Is the concept of 1/4 accurately visualized?  Will sloppy drawing confuse the issue of equivalent fractions?

PRO:  I’m in favor of any system which makes collecting and using details from word problems a major focus of study.  Personally, I like “just working problems,” but I haven’t found many students who share the feeling.  The current emphasis in STEM study is the usability of skills, requiring practical applications presented through word problems.  Singapore Math certainly places a premium on word problems and THAT can be a major plus.

CON:  I am not convinced, however, that the time required for accurate artistic renderings is worthwhile in the upper grades.  I’m positive I’d never use the system on a timed test like the ACT.  The statistics which suggest that Singapore’s educational system for teaching math is superior only measures performance at the fourth and eighth grade levels, and at some point, students will have to abandon the visual representations in favor of international conventions. I’m eager to observe how the transition from Singapore Math to traditional solutions will be made.

PRO:  BAR modeling could prove to be a useful key for helping VISUAL LEARNERS to develop an intrinsic understanding of fractions and to translate the details of some word problems.  Singapore Math offers opportunities for revisions that could inspire MANUAL LEARNERS by using clay, for example, and cutting 3-d rectangular prisms into subdivisions.  Innovative teachers might experiment with modifications that are so successful that the next new approach could be called “American Math.”  For the time being, I plan to get as much traction from the latest new system as I can.