Curriculum Review: Singapore Math (Grades 1-3)
Singapore Math
Published by the Ministry of Education, Singapore
Available in this country from
Sonlight Curriculum, Ltd.
8042 South Grant Way
Littleton, CO 80122-2705
FAX (303) 730-6292
Phone (303)730-6292,
and from
SingaporeMath.com, Inc.
19363 Willamette Drive #237
West Linn, OR 97068
FAX (503)722-5671
Phone (503)727-5473
NOTE. I have gotten more angry e-mail about remarks I’ve made about Singapore Math than on any other subject, including issues of creation and evolution. Those remarks were often quoted out of context or inaccurately. PLEASE DO NOT EXCERPT THIS REVIEW IN ANY FASHION. PLEASE DO NOT DISTRIBUTE IT TO E-MAIL LISTS. If you want to share it with others, give them the web address so that they can read it in its entirety.
The newest math option for classical home schoolers is Singapore Math, a curriculum originally used in Singapore, where math students score more highly than students from other countries in internationally administered tests. Because I was unable to get review copies from the Ministry of Education in Singapore, I bought the Singapore program myself. I could only afford to buy the first three years (although I hope to invest in the other books later). My impression from the Singapore scope and sequence is that the primary books are all similar in layout and content, so my remarks on grades 1-3 are probably valid for the program through grade 6. I have not yet seen the 7-12 materials.Program description
Each semester of the Singapore Primary Math program consists of one course book and two workbooks (all in paperback). For example, the first grade program consists of Course Book 1A and 1B (first and second semester) along with Workbooks 1A, Part One, 1A, Part Two, 1B, Part One, and 1B, Part Two.
Singapore’s attraction is its focus on teaching mathematical thinking, as opposed to rote problem solving. For example, one of the earliest books (1A) begins to assign “mental math” puzzles as soon as children learn to count. They practice counting from 1-10 and then backwards from 10-1, and are immediately given a problem to solve: a picture of a ferris wheel with six “chairs,” each chair containing a sequence of three numbers with one number left blank:
___, 8, 9
3, __, 1
__, 6, 5
2, 1, __
5, __, 3
10, 9, __The first-grader has to decide whether the sequence runs forward or backward, and then decide which number should fill in the blank. A workbook exercise, meant to follow this course book exercise, reinforces the same skill. This is a very different approach from that found in more traditional American math programs such as A Beka or Saxon, which tend to assign the child a single task (counting backwards) and repeat it a number of times before asking the child to use that task to solve a problem (filling in number sequences). In the Singapore math program, children are taught to use new skills immediately in the solving of problems.
Because the goal of the Singapore program is to produce an understanding of the way mathematical processes work, skill are introduced differently than in American programs. Multiplication and division are begun very early (right at the beginning of second grade), so that the student is aware of the relationship between multiplication/addition and subtraction/division.
The books are colorful, with cartoon-like illustrations and pictures showing each new concept worked out with actual objects (very important for grammar-stage students). The accompanying workbooks are meant for the child to write in.Be aware…
Singapore doesn’t do American measurements, weights, or money, so you’ll have to supplement this part of the program with an American program (Miquon is affordable).
Singapore does not come with manipulatives. I think that many of the problems would benefit from some hands-on working out, so I suggest that you invest in the Common Sense Press “Math Bag,” affordable snap-on blocks (a little like Legos) arranged in sets of 1-10 plus a base.
In Singapore, this program is only part of the math program that produces high-scoring math students. After-school enrichment classes that drill math facts are also part of the Singapore system. If you want to get the same results that Singaporean teachers get, you MUST buy and USE a math drill program in order to reinforce knowledge of math facts. As it is sold in the USA, the program has no math-facts practice and very little repetition. It is important that students understand why multiplication, addition, division, and subtraction work, but it is equally important that they go into higher mathematics with an automatic knowledge of these math facts.Pros and cons
Singapore Math is geared towards producing mathematical thinkers, and it does this very well by walking children through all the component parts of a problem and then presenting them with the whole problem to solve. In this way, it teaches them to think through the different steps of a problem rather than trying to plug it into a formula. The difference between the Singapore and Saxon methods seems to be that Singapore introduces the problem solving earlier. Above, I described a Singapore problem where the child counts backwards and then immediately is asked to solve a number of sequence problems where she has to figure out a two-step problem solving procedure: first, identifying the sequence as either counting forward or counting backwards; second, filling in the correct number to complete the sequence. Although this is a simple example, it is very typical of the Singapore method. From the earliest lessons, students have to put their new knowledge to use in a problem that requires them to do more than one step at once.
By contrast, when Saxon introduces counting backwards in Lesson 34 of Math 1, the child practices counting backwards at the beginning of every lesson from then on. (Singapore doesn’t return to the skill at all). Counting backwards is never used to solve a problem. Rather, it lays the foundation for the child to be able to count backwards by 10s (introduced in lesson 52); this counting-backwards-by-tens skill is then reviewed for every lesson until it is used as a way to explain subtracting in the tens column (lesson 119). Saxon does not put the “parts” of a problem together until each separate part has been extensively drilled.
Singapore’s great strength, then, is also its weakness. From the very beginning, children who use Singapore Math are taught that everything they learn can be used (right away!) to solve a problem. This develops the habit of looking for solutions, rather than groping for formulas.
On the other hand, Singapore’s method of immediately using a new skill for problem solving is inevitably going to frustrate some children. Especially in the early years, many children NEED rote repetition of skills BEFORE they are able to put those skills into use. For these children, Singapore’s method will not be as effective as Saxon’s. Although the classical model is not intended as a straitjacket, it does suggest that most young children are not yet capable of thinking critically and/or abstractly. In grades 1-4, most children learn better if their lessons are based on concrete facts and examples; although some children are able to extrapolate from concrete facts and examples to more abstract ideas, many elementary students need time to mature before they are able to do this. (For example, if you tell a second grader a story about a cruel tyrant, he will understand clearly that this king was unjust. But he will generally not be able to answer the question, “What is justice?”) The Singapore method leads children into “logic stage” thinking much earlier than other programs. If you try Singapore and your child is frustrated, this probably signals a maturity problem; set the program aside and come back to it in a year.
Furthermore, although I am very enthusiastic about the way Singapore teaches mathematical processes, I am not very happy with the long lag time between the introduction of a new skill and its repetition. For example, a student who does Singapore 2A learns addition, subtraction, multiplication, and division. 2B reviews all of these in the first 35 pages, but then devotes the rest of the book to money skills, telling time, capacity, graphs, geometry, and area. The book does supply review problems in the skills learned earlier, but the lessons themselves move completely away from addition, etc. At the beginning of 3A, the skills are reviewed very briefly – too briefly for most students, who will need to look back at 2A to remember what they were doing and why. (This problem is compounded if you take the summer off! That’s a long time between multiplication lessons.) You will need to revisit the lessons on multiplication and division occasionally in order to make sure that young children don’t forget how and why to perform the skill. You’ll also need to improvise some practice. (For example, the counting-backwards lesson provides the child with three opportunities to count backwards in the lesson, and one workbook exercise. Most children are going to need a lot more practice than this – and a lot less practice than the 92 repetitions suggested by Saxon.)
Generally, I could wish that the Singapore materials provided a little more explanation when introducing new skills. The explanations given are generally clear, but not lengthy; if a certain skill doesn’t “click” with the student, you will have to think of a new way to present the information yourself. Also, the explanations are often given in pictures rather than in words, which may bother some parents who would prefer to have written explanations on how to teach the concept. (Incidentally, the Sonlight description of Singapore Math says, “Some students who are not good at math have problems with what they perceive as a lack of adequate review and drill.” Although I generally have a high opinion of Sonlight, I think this is unfair. MANY students need clear and detailed review and drill in order to keep math skills current and sharp; it’s not helpful to label these students as “not good at math.” Any child can be “good at math” when matched to right math program.)
There are also serious difficulties with the Saxon approach. Children who drill, drill, drill skills before ever using them often begin to view math as a set of rote facts which they should spit back out on command; they turn off the “reasoning” portion of their brain, and it is very difficult to turn it back on. Some students who do Saxon for the first four grades will be unable to make the switch over the “logic-stage” thinking when they enter the middle grades; they will have increasing difficulty as mathematics grows more abstract, and Saxon provides minimal instruction and practice in mental math. The Singapore approach provides a much clearer entry into logic-stage thinking and much better “logic-stage” work.
A final concern about Singapore: There are no (and I do mean no) teacher aids. The only teaching help you get in the early grades are the pictures in the course books. Many parents will do just fine with this, but it is very different from the Saxon approach (which gives parents a script to read in the first years). For example, the Saxon lesson on halves and fourths begins, “Today you will learn how to divide a solid into halves, fourths, and eighths. Today we will use apples for our math lesson. [Cut apple A in half vertically, diagram provided.] “When I cut the apple like this, I am cutting it in half. How many pieces do we have? What do we call each piece? One half. How many halves are in one whole? Two. I will cut each half of the apple in half again…” and so on. Naming fractions (using the words “one half, one fourth, one eighth”) is done in a later lesson. Writing fractions (1/2, ¼, 1/8) occupies yet another lesson. Each time the parent is given suggested dialogue.
When Singapore Math introduces fractions, the parent gets a picture of a whole pizza, a pizza divided into halves, and a pizza divided into quarters. The text says, “Divide a circle into 2 equal parts. Each part is a half circle. Divide a circle into 4 equal parts. Each part is a quarter circle. Which is greater, ½ or ¼?” The parent has to explain in her own words that one half = half of something, that it is written with a 1, a line, and a 2, and why that particular notation represents a fraction. No further explanations are given. (There is, however, an online forum hosted by SingporeMath.com.)
Math-U-See, which is designed for home schoolers, uses many teaching methods that are similar to Singapore’s, but provides teaching videos; if you like the method but feel unsure of your ability to teach without aid, consider this program instead.Final thoughts and recommendations
I have long been unhappy with the state of math and science in classical education. History and literature resources are everywhere, but too many classical educators downplay (or ignore) the important of mathematics. In a true classical curriculum, mathematics is taught as a language which must be understood and spoken fluently.
However, many parents (most of us?) don’t speak fluent “math” and aren’t able to lead our children where we haven’t yet gone ourselves. How can we change this?
I don’t think that the answer lies in any single math program – any more than Powerglide alone can turn you into a fluent Spanish speaker. You will only learn to speak Spanish fluently if you carry on conversations with several different Spanish speakers who use different idioms, vocabulary and accents. In the same way, a person who is considered literate in English has read not just one book, but a multitude of books that present different arguments and use different vocabulary and sentence structure.
In my opinion, the best way to do math classically is to use more than one program simultaneously. Each math program uses different mental “tricks” and procedures to teach mathematical concepts, and each program has its flaws.
Consider, for example, the way that Singapore teaches students to add two digit numbers with “carrying” or “renaming”:2A, p. 13 Teaches the concept of hundreds, tens, and ones (place value)
2A, p. 22 Teaches the concept of addition
2A, p. 25 Teaches two-digit number addition: add tens and then add ones.2A, p. 36 Teaches “addition with renaming.”This exercise is done with pictures (bundles of pencils) and covers only “addition without renaming” (for example, adding 236 + 362, so that no amount in the hundreds, tens, or ones columns exceeds 9).2B does not cover addition at all (a weakness in the program: see above)This exercise explains with pictures that when the ones in the ones column number ten or more, ten of the ones have to be moved over into the tens column, which changes the number of “tens.” (Also covers moving tens into the one hundreds column.)
3A, p. 8 Reviews the concept of hundreds, tens, and ones, and adds thousands
3A, p. 19 Gives the student an addition problem that involves “renaming”3A, p. 24 Explains the concept of renaming or “carrying” againThis exercises begins with 90 + 54, but doesn't review the concept of renaming (I would think that most children would need to turn back to 2A to remember what they’re doing).
3A, p. 36 Introduces word problems involving carrying and renaming
3B, p. 8 Now that the concept has been explained, Singapore gives the student a mental “trick”Saxon takes a different path:The "trick": when you add two-digit numbers, first add the ten of the second number to the first number, and then add the ones of the second number to the result (so if you’re adding 56 and 29, first add 56 + 20 = 76, and then add 76 + 9 = 85).Math 2, Lesson 28 Teaches counting dimes and pennies (introduction to place value)
Math 2, Lesson 35 Teaches adding tens to numbers (increasing the tens column)
Math 2, Lesson 38 Reviews place value using dimes and pennies
Math 2, Lesson 43 Teaches trading pennies for dimes (review of relationship between ones and tens
Math 2, Lesson 47 Reviews adding tens to numbers (increasing the tens column)
Math 2, Lessons 66-67 Adding two digit numbers without “carrying” or renaming, using dimes and pennies
Math 2, Lesson 71-72 Adding two digit numbers with renaming (here called “trading”), using dimes and pennies
Math 2, Lesson 85 Adding three two-digit numbers, using money
Math 3, Lesson 15 Reviews adding ten to a two-digit number
Math 3, Lesson 23 Reviews renaming or “regrouping” tens and ones, using dimes and pennies
Math 3, Lesson 69 Now that the concept has been explained, Saxon gives the student a mental “trick.”Which method is better? In my opinion, the one which the student understands most clearly. In both cases, it is possible for the student to learn the mental trick without thoroughly understanding why it works, although the sheer amount of repetition in the Saxon method makes it easier for this lack of understanding to escape detection. But the strongest mathematical training of all would come from a combination of programs – in which the student is taught to do a mathematical process using several different methods and mental procedures.The "trick": when you add two-digit numbers, first add the tens and then adds the ones (so if you’re adding 56 + 29, first add 50 + 20 = 70, then add 6 + 9 = 15, and finally add 70 + 15 = 85).
Currently, Singapore and Math-U-See are “thought-oriented” math programs available to home school parents; Saxon and A Beka are “skill-oriented” programs. A combination of Saxon + Singapore, or Saxon + Math-U-See, or Singapore + A Beka, or A Beka + Math-U-See, may come closest to fulfilling the goals of classical education. Math-U-See + Singapore would also be an excellent combination, as long as you use MUS’s supplementary drill sheets. Treat one program as primary and the other as secondary; when you cover a concept in the primary program, look it up in the secondary program and see whether it is explained and illustrated differently.One final remark: Singapore and the SATs
I have not personally seen the 7-12 books, but the Singapore scope and sequence certainly seems to cover all the topics tested on the SAT. However, the presentation of the Singapore material is very different from that found in American mathematics programs. While this is a plus for many parents who are frustrated with the level of math achievement in America, it does present a complication; a student who has done Singapore algebra and geometry exclusively will not be familiar with the some of the types of problems found on the SAT. These problems are modelled after those found in American mathematics programs.
Keep in mind that the SAT quantitative section is not (in my opinion) a measure of how well a child thinks mathematically. It measure how well a student has learned a specific set of information. If there is a mismatch between the format of the child’s math program and the information tested on the SATs, the child can end up with a test score that doesn't match his actual ability.
Because the SAT is such an important “leveller” for home school students (showing that they have reached at least the same level as their classroom-schooled counterparts), parents would do well to take this possible mismatch seriously. (I use the word possible because Singapore math has not been used in this country long enough to yield any useful statistics related to the SAT; anyone who tells you that Singapore is either wonderful or awful for SAT scores is merely guessing. My remarks here fall under the category of “educated guess.”)
Singapore does teach students to “think mathematically.” Unfortunately, this isn’t all they need to do well on the SAT. Keep these facts in mind: Studying directly to the SAT (using an SAT prep book) raises test scores -- especially if the studying is done in the weeks immediately before the test is given. It's difficult to see why this should be, unless the test is testing content which can be quickly assimilated (and probably as quickly forgotten), rather than primarily testing mathematical thinking. Also, it has been well documented by the College Board (and endlessly discussed in the Chronicle of Higher Education -- see their website for unending articles on this topic) that minority students and students from different cultures do not score as highly on any section of the SAT as white middle-class American students. The only reason anyone can suggest for this is that the SATs assume a certain "body" of cultural knowledge, and that students who don't have it don't test as well. Again, this would not be true if the SATs tested for clear thinking rather than a body of information. This particular statistic really does make me cautious about untested programs from different countries -- it is certainly possible that there might be a cultural-difference problem.
Recommendation? High school students using Singapore should begin, in eighth grade at the latest, to work through an SAT prep book from Barrons, Princeton, or ETS. Ideally, they should do this twice a week until the SATs have been taken. (At this rate they should be able to work through several different prep books – which is the strongest preparation of all.) This will cover any potential gaps between the Singapore program and the American programs which the SAT is designed to match.
Parent Comments
Add your comment: e-mail the Webmaster!How I wish I'd seen this review in September. I could have saved my kids countless hours of frustration. Two of my children started in NEM 1 this year after 6 and 7 years with Saxon. Both are A and B students, probably a B in Math. With the switch to Singapore they became C and D students. Granted, Singapore stretches thinking skills. But this must be done by a competent teacher. Evidently they do all sorts of supplements even in the logic and rhetoric stages in Singapore, so merely buying the book doesn't cut it; it's a whole new way of thinking and difficult to grab onto in junior high. You are right, 2 methods are needed if you want to use Singapore at all, but I'm wondering if it is really worth it. They do miss basic concepts, there isn't adequate review, and the students don't have enough examples and explanation in the books to be able to figure out problems themselves! It is very teacher dependent. Perhaps it works in Singapore because the teachers have better training, it starts right away, and there are
supplements.-- daggfamily@juno.com It is my understanding that the teacher's guides for Primary 1 are now available and that Jenny from the Singapore.com site is now writing manuals for homeschoolers for levels 2 & 3. (These homeschool manuals will be available at Sonlight.)
-- sspille@hot.rr.com I appreciate you putting the review up on the Web where a wide variety of
people can see it. My comment on the review is that the materials for later
years in the Singapore sequence, which the reviewer acknowledged not seeing,
are indeed more than adequate preparation for a test like the SAT I:
Reasoning Test math section. A student who stays on age-grade level in the
Singapore program will not have a problem on the SAT. The careful
interrelation between and extension of previously learned facts and concepts
throughout the Singapore Math program ensures that children who work through
it gain a DEEP understanding of math that can readily be applied to
unfamiliar situations. Thus far my oldest son has found math tests, even
"out of level" math tests he has taken for placement in gifted programs, to
be cakewalks after taking the Singapore curriculum (following earlier use of
the Miquon Math program) through book 4A.As a parent advising other parents, I have to figure that I may differ in
some substantial way from other parents trying out homeschooling math
programs. I have noted with interest on the Sonlight math forum, where
multiple programs are discussed, that a great variety of parents, some with
very limited backgrounds in mathematics, have nonetheless been glad to make
the switch to Singapore Math. That switch seems to be happier the earlier it
is made (perhaps because many "drill and kill" math programs destroy the
curiosity almost all children have about mathematics in early childhood).
Switching to the Singapore program also seems to be easier the more the
parent reminds the child that initially there will be some learning of a
more broad and thoughtful approach to math to tackle before zipping through
the curriculum itself. But children catch up and get back on grade level
once they practice the unfamiliar aspects of the Singapore program,
according to the parent comments I have read on-line.The international data from the 1999 TIMSS study suggests that the strength
of Singapore in international comparisons derives largely from how well the
BELOW-AVERAGE students do compared to students in other countries (the
bottom students in Singapore know math as well as average students in the
United States, according to the TIMSS testing), so the implication is that
the Singapore program is, at the very least, not harmful to students who are
low in the ability range among children. I do not disagree with the reviewer
about second-sourcing math programs: every parent ought to use multiple,
differing materials for homeschool study of math. I would simply gently note
that I rather think that Saxon Math is in more serious need of
second-sourcing than Singapore Math.Thanks for sharing and for inviting comments from readers.
Karl M. Bunday "pray for us" 2 Thessalonians 3:1
Learn in Freedom (TM) Web site http://learninfreedom.org
kmbunday@earthlink.net (preferred address)
My second grader has shown a great deal of improvement as well as interest in math since I switched to Singapore math this year. Not only is he not "bored" with math, but his critical thinking skills are wonderful.
The company I ordered the curriculum from offers a teacher's guide that gives ideas on how to present the math concepts to the student. At first I was a little bit unsure that my son would understand since we had used Alpha Omega and Bob Jones in the past. Singapore math seems to make a lot more sense to him. There are now supplemental math books to use for practicing the math skills learned.
So far my experience with Singapore math as been a pleasant one. I understand it is not for everyone, but like all new things it takes time to develop. When I first looked into this curriculum they did not offer the supplemental books, but now they do and I have been amazed at how much my son has learned.
Mrs. K. Burton
BURT4JC@msn.com
Thank you so much for your very insightful review
of the Singapore math. I have used Earlybird, 1A/1B,
3A/3B, 4A/4B. It definitely taught my daughters to
'think outside of the box'. However, my youngest exhibited
the weakness I feel is in Singapore - trouble with basic math
facts. (add/subt.) Even though I used their review materials,
my daughters would forget things. Much of the review is
by chapter. They do not carry it through the program like Saxon
does. I used Saxon in the younger years & did not like it. BUT,
my daughter had her math facts down cold, by golly! Your suggestion
to use a combination of programs is an excellent one. I do miss the
way Saxon keeps reinforcing the facts. I love Singapore's word problems
& mind-stretching approaches to solving problems. I think to use Saxon
as the main program & Singapore for its creative problem solving would
be a great blend. They complement one another very nicely.
I do not think your review of Singapore was unfair at all. It was "fair and
balanced". We homeschoolers get so sensitive about the programs we
like.
Thank you again.
LBweaves@cs.com
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