SINGAPORE MATH: AT-HOME RESOURCES

WARNING:  THERE IS MORE TO SINGAPORE MATH THAN THE CLASSROOM

Even critics of the Singapore Math program could not object to the proclaimed objectives of teaching students to the mastery level, using real world situations for problem solving, and encouraging communication in math.  After all, these same goals have been recommended by the National Council of Teachers of Math for decades.  Everyone interested in math wants our children to love math, not just tolerate it through 11 or 12 years of compulsory education.

And even the critics may not call enough attention to the need for out-of-school support if students are to achieve that mastery of basic arithmetic concepts.  The Singapore Math classroom curriculum stresses conceptual understanding to the delight of advocates, but procedural and skills proficiencies are often the under-reported secondary requirements that critics demand and that are an integral part of Singapore's education program. 

Unlike the American system in which “school” ends at 2:30 and is replaced by dance class, sports, and other extracurricular activities, in the country which inspired Singapore Math, nearly every student receives supplementary homework assignments, as well as additional rehearsal through math clubs and “after school tuition classes” like our tutoring services.  The reported 1995 (http://nces.ed.gov/pubs99/1999081.pdf) through 2007 (http://nces.ed.gov/timss/table07_1.asp) math success in Singapore may be more heavily dependent on the cultural significance placed on education than we realize.

With this possibility as our hypothesis, effective implementation of Singapore Math in American schools will require at least moderate changes to our priorities.  It will be necessary that after school activities devote more time and effort to the repetitive work -- those dreaded “math facts” like multiplication tables.

Practice in basic computation can be tedious for both parent and student.  Multiple worksheets with 25, 50, 100 problems are boring, to be generous.  A parent verbally making up sample problems is demanding.  The chances are that none of us is going to follow through long enough for our kids to gain the level of skill that we would like.  We need a helper.

Tutoring Resources uses a game we call “War on Integers” which is played like the familiar card game War.  Contact us at www.tutoring-resources.com for a free copy of instructions for playing with addition, subtraction, multiplication, or division.  The game encourages the student to practice in a fun way that doesn’t depend on age or grade level to “win” and promotes the much needed rehearsal.

The internet is another readily available resource.  There are many sites that provide colorful, entertaining games for students from kindergarten through sixth grade and even higher.  My personal favorite (because I played it with my own kids back in the day) is NUMBER MUNCHERS.  I have it on floppy disk (I did mention that it was “back in the day”) but for today’s electronically-literate youth, it’s available for iTunes, iPhone, and iPad Touch.

Other sites to explore include
www.wallofgame.com  (Number Munchers is available on this site.)

www.Sheppardssoftware.com  (A listing of several useful sites.)

www.IXL.com  (This one makes it easy to access appropriate games by grade level and even goes through Algebra.)

www.Cool.math-games.com (This site may not be as distracting for kids who are accustomed to websites as it was for me, but I would not recommend it for anyone with focus and concentration issues.)

As a general rule, games will require dexterity with either arrows or a mouse.  I found them all easier to play on a desktop than laptop computer.

Want more suggestions?  Just search the web for ‘math games’ and be prepared to spend considerable time checking out a myriad of sites that might appeal to the specific interests of your student -- and playing a few yourself.  I lost more than a few minutes on the pop-up www.ahhhhh.com, which is apparently a soft drink website ad, as mezmerizing as it is subliminal.

If our kids are going to get the promised rewards from Singapore Math, parents need to make it a point to provide the supplements that will fill in the gaps.  Luckily, plenty of resources are available to make our job easier. 

SINGAPORE MATH: PARENTAL INVOLVEMENT

So your school district is all in a lather over Singapore Math.  Some are dead set against the changes coming in the Fall and others are so enamored that they seem almost manic.  This is a common scenario:  two sides, both at the extremes.  Critics and advocates alike will implement the new methods, procedures, and texts like scientists conducting an experiment.  The classroom is the lab.  Results will be evaluated somewhere down the line and assessments made about the effectiveness of the program.  It has happened before with “new math,” Everyday Math, and many other attempts at improvement that have resulted in both success and failure. 

But YOU are the parent and what happens to, for, and with your child is a permanent fixture of his or her education with important future ramifications. To YOU this “experiment” MUST WORK.  Supporters of Singapore Math do not hesitate to stress the importance of parental support for successful implementation.  Some suggest that after-school tutorials and homework are integral factors in the reported success of elementary students in Singapore.  If you are among the group of the hesitant, unsure of your own math knowledge and ability to adjust to Singapore Math, this article will help you become a competent homework assistant.



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TIPS FOR PARENTS NEW TO SINGAPORE MATH

1.  Many of the “informational” publications about SM are lean on specifics.  You’ll hear that the Singapore Math program as amended for American Schools teaches mathematical concepts from concrete through pictorial to abstract, devoting more time to fewer topics and stressing “mastery.”  This limited information seems to me to be more of a sales pitch than useful data. 

It is up to the parent to drill down to details that will help each student to learn the math fundamentals.

For example, here is the sequence of study for the first week or two of each grade:
    Kindergarten:      Numbers to 5
    First Grade:         Numbers to 10
    Second:              Numbers to 1,000
    Third:                  Numbers to 10,000
    Fourth:                Place Value of Whole Numbers
    Fifth:                   Whole Numbers
    Sixth:                  Positive Numbers/Number Line

With this information in mind, you can help prepare your student for the first few days of school by reviewing the numbers that will be covered.

2.  To ensure that you are in the loop throughout the school year, be sure you are receiving the “School-to-Home Connections” newsletters from the teacher at the beginning of each chapter.  These include a summary of the upcoming lessons, vocabulary, and an activity to support learning at home.  The newsletters are available to the teacher as prewritten resources which can be printed for general distribution, emailed, or posted on the school’s homework website. (Sixth through eighth grade newsletters are called “Family Letters.”)

3.  Many of the “new techniques” used in SM are similar to systems used in the 1950s and 1960s.  That’s a little before the grammar school dates of most of today’s parents, but here is a tremendous opportunity to get gramma and grampa involved.  Ask them about “friendly numbers.”

I am of that era and remember breaking large numbers into workable groups. (In fact, I still use this system for mental math, which is a chore for me rather than a talent.) For example, studies have shown that most people can “picture” numbers up to 6 in patterns found on playing cards.  The number 7 becomes more difficult to envision, but it can be broken up into 5 and 2.  Adding 5 and 7 is the same as adding 5 plus 5 plus 2.  Adding 5s is a familiar pattern; 5 + 5 = 10.  Add another 2 to find the sum of 5  + 7.  In Singapore Math, this is called “adding with regrouping.” Learning the process is facilitated by using manipulatives.

Subtracting with regrouping is similar.   Seven can also be broken up into 3 and 4.  So 13 - 7 becomes 13 - 3 to get 10 and then subtracting 10 - 4.  Ten is the friendly number in this case.

For early elementary students, using manipulatives to find number patterns (called number families in some very old textbooks) is a common practice.**

**  Manipulative “dots” are available free of charge to fans of this blog.  Become a fan and send a request for a set of 100 Dots through the comments section that follows this article or email tutoring.resources@yahoo.com

4.  Singapore Math uses “place value mats” as early as first grade.  The classroom teacher should have a template which can be copied and distributed to families, or you can easily make your own by replicating the following pattern.  Columns should be long enough to accommodate 5 manipulative dots (breaking 10 up into friendly 5s).  Laminating the mat will ensure that it holds up to 6 years of single student use or as a sibling hand-me-down.  At the second grade level you’ll want headings for thousands, third grade ten thousands.

5.  ADDITION (OR SUBTRACTION) WITH RENAMING are the SM terms for “carry the one”  and “borrow” with which you might be more familiar.  Be sure to review the vocabulary from the School-to-Home newsletter and ask the teacher if the connection to the math you know is not immediately obvious.

6.  After manipulatives, SM might move to drawings to transfer tactile knowledge to the more conceptual/visual format.  You may want to practice setting up word problems using this system.  It is not complicated when you've mastered the routine steps for creating bar models.  Check out The Normal Genius blog, "Singapore Math: Using Bar Models," posted on August 2, 2013.

7.  Other techniques that SM uses to engage students and stimulate higher level thinking include a heavy use of word problems (called “practical applications” in some older textbooks and college courses) and classroom discussions.  The format frequently presents a story that would use some form of math to solve a problem, let the students work independently or in groups to “think it out,” followed by presentations of their solutions.  This practice has long been supported by the National Council of Teachers of Math which emphasizes both real life situation applications and the importance of communication in mathematics either through speaking or writing.  It encourages students to identify and recognize their own problem solving thinking, a metacognitive element of learning.

You can help your student prepare for this task by practicing explaining the steps to a solution for sample homework problems.  Your own ability to model the process could be a valuable asset for your student.

8.  Thinking through problems is only part of the SM formula.  Rehearsal of basic math facts is still required but will need to take place at home.  Ask the teacher for supplemental materials outside of the workbook exercises.  Try chanting the multiplication tables (like you did to learn them) while throwing a ball back and forth or bouncing it on the floor for students who need to move while learning.  Or chant to a particular beat for the musically inclined.  Use your imagination to design drills that meet the specific needs of your unique student.

9.  Not necessarily part of the Singapore Math program, but a useful tool for upper elementary students when word problems can become quite complicated, is a strategy I call “Is that your final answer?”  It sets the framework for what the answer will look like, set up at the beginning of the problem solving process and insuring that the student doesn’t stop too soon or go down a blind alley while working out a solution.  In either math symbols (4X = ___ ) or words (4 pencils cost ___ ), the practice is similar to checking the question after working through the math.

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So your school district is all in a lather over Singapore Math.  But you are calm, cool, and collected because you are prepared to help your student make the most of the program and maybe even gain confidence in your own math skills at the same time.

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.

The traditional solution looks like....

     X = total lemonade
     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

IS SINGAPORE MATH WORTH THE HYPE? 

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. 

If you find interesting problems from your student's class, please share them in the comments.  

The greater the number of unique examples, 

the deeper the understanding.

NUMERACY: A FIRST STEP TOWARD PRESCHOOL MATHEMATICS

This is a special article on the blog, just for parents of all those youngsters I met at Northbrook Days and who I hope will become Tutoring Resources students in 8 years or so from now. Usually I’m totally focused on the preteen to adult ages. I like working with kids old enough to question a teacher, even challenge the teacher’s knowledge.

But Math education starts long before Middle School and parents are the first and most influential teachers that every child can have. So here are a few things to do before students get into Preschool.

NUMERACY: This is the internal understanding of numbers. It involves number order, recognition of number symbols, and the one-to-one relationship of counting. Using the Tutoring Resources Math Cards distributed at our booth, select a few of the positive numbers, say 1 through 5 for the youngest kids and up to 10 as a child progresses. Lay them out on the table in numeric order.

Activity #1: (for ages as young as 2) Point to each number in order and say it’s name. Show the child how to point to the letters as you say their names. Ask the child to repeat the numbers after you’ve said them, starting with one at a time and progressing to the whole list at once. Eventually, have the student name the cards without help.

Activity #2: Using the same numbers that have been practiced in Activity #1, ask the child to point out a certain number, first with numbers laid out in order and then with cards randomly spread on the table.

Activity #3: Using manipulatives (navy beans, candy pieces, dots....something large enough that the child can pick it up without much discomfort), demonstrate the concept of each number by laying one item on the #1 card, 2 items on the #2 card, etc. As a second step, put the manipulatives in one central location (like a dish) and demonstrate that each number is the previous number with one more added. Help the child “count” with you, identify the number verbally and pick out the card with that number on it. Evenutally, have the child lay out the right amount of manipulatives to correspond with a random number that you pick from your deck.

Activity #4: Start “counting” stuff.....lots of stuff. Count the number of chairs at the dinner table, the number of plates or forks while setting the table, the number of shoes in daddy’s closet, the number of winter coats in the front hall, the number of toy cars or stuffed animals or Barbie doll dresses. Count, count, count until it becomes “natural to count” 1-2-3... These are the “natural counting numbers” that the student will use in kindergarten and first grade and which are the first group of “Real Numbers” as described in Pre-algebra. Imagine! Your 3 year old is already learning one of the Pre-algebra concepts!!

OTHER RESOURCES: The library has numerous number books with themes that may be stimulating to your child -- animals, characters, etc. There are puzzles that reinforce the shapes of numbers; I just found one at the one-dollar store.

Repetition and exposure are the keys to teaching NUMERACY, the vitally important first step in the understanding and love of math. It is my goal that your child have FUN with numbers and develop a sense of accomplishment and competence that will spur him or her to success in Math class. To receive a packet containing the Tutoring Resources Math Cards, dice, dots, and sample games for first grade through sixth, contact your local Tutoring Resources center in Barrington or Northbrook. Visit the website at
www.tutoring-resources.com for contact information.