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A Singapore Math Adventure

by Jenny Hoerst

I first heard about "Singapore Math" five years ago on one of the homeschooling forums. At that time, there were only a few comments here and there about this "new" math program. I was in the ultimate search for the perfect math program for my children, or at least a better math program, having gone through a variety of the popular programs. None of these really satisfied me in how they were teaching my kids (or not teaching them, as seemed to be the case) to think mathematically. I visited the website (which at that time had little concrete information) and ordered a few levels corresponding to my kidsí grade levels. I was a bit skeptical, partly because there were no answer keys at that time, but the books were inexpensive enough to experiment with.

I must admit that when I first got them (I ordered 3A, 6A, and the first secondary level text), I was disappointed by their appearance Ė the Primary Mathematics books were thin and looked from the outside like workbooks I could get in the grocery store. However, the secondary book, supposedly for 7th graders, looked quite good ó a 7th grader would need to have had good preparation for it. So I carefully looked through the Primary Mathematics levels I had and was pleasantly surprised at the types of questions they had. These were not simple, run-of-the mill, boring questions ó I had to actually think to answer some of them

I soon realized that, though a grade ahead of the levels I bought, my kids were not prepared for the challenge in these thin, unprepossessing books. So I ordered earlier levels, then other levels, and studied them, and actually enjoyed learning from them. I recognized the strength of this program enough that I wanted to make it more accessible to other homeschoolers, who would hesitate to use it because of the lack of teacher helps. So now I work for as their curriculum advisor. On the side, I have written a number of Home Instructor Guides, Solutions Manuals, and have answered millions of questions from other homeschoolers on the companyís forum and by email. My kids sometimes complain that I am more ready to answer questions about math from other parents than from them!

Somewhere along the line I even taught my own kids with these books. I had to work ahead of my daughter in the secondary level books but I rather enjoyed the challenge, and both my kids and I enjoyed the feeling of accomplishment when we actually arrived at the correct answer. Though my kids didnít have quite the same enthusiasm as I had, after all, math was schoolwork, and schoolwork was something that got in the way of learning, I think they actually enjoyed doing problems that were not always just "practice what you have been taught," but ones from which they actually learned as they did them. My daughter -- not a "math" person -- even objected when I tried to switch her to something easier. For my younger two boys, I added in some of the supplementary books, and I know the problem-solving skills my kids learned from all of these books have held them in good stead. My son could often come up with a different, more efficient way of solving some of the problems than I could. He recently took a placement test at the community college after finishing the 8th grade text and placed into pre-calculus. He said there were things he hadnít seen before on that test, but he could "figure them out."

I think this is the biggest strength of this curriculum ó it gives students the ability to "figure things out", that is, to solve problems they havenít seen before, problems where they havenít already been led step-by-step through the solution of a very similar problem. Students are taught mathematical concepts, not expected to rediscover them on their own. However, they are expected to apply these concepts quickly to new situations. It will become obvious quite soon whether the student does not have an adequate grasp of the concepts ó there is little chance of a student skating along by simply following procedures by rote. Many of the problems in this curriculum are well thought out to advance and deepen the studentís understanding of the concept, rather than simply practice a process, and if a student struggles through a problem, it can and should be thoroughly discussed, and perhaps approached from different angles, to enable the student to grasp a better understanding of the concepts required to solve the problem.

For example, a child in the Primary Mathematics 1 is taught (or reminded) how to count to ten, to understand the relationship between numbers in a "number bond" or a part-part-whole context (5, 3 and 8 make a number bond), to see addition not only as "adding on" but also as finding a missing whole, and to see subtraction not only as "taking away" but also as finding a missing part. Then, the learner is taught numbers to 20 in terms of place-value, and how to apply this knowledge along with addition and subtraction through ten to addition and subtraction through 20 Ė before memorizing the math facts through 20. The concept of part-whole is carried throughout the primary level, and in particular is applied to word problems so that the student understands what operation to use when interpreting word problems. A student in the Primary Mathematics 3 can apply the part-whole concept to equal parts, or units, in multiplication or division, and in Pri-mary Mathematics 4, he or she can apply equal parts to fractions and can solve word problems involving fractions by finding the unit in the same way as finding the value of a unit with division. This concept, in turn, can be applied to ratios in Primary Mathematics 5 and to percentage problems in Primary Mathematics 6. The unit, of course, will eventually relate to the algebraic "variable".

Another major strength in the Primary Mathematics is the use of "bar diagrams" to solve word problems. Different quantities are represented by rectangles whose relative lengths reflect the relative values of the quantities given in the word problem. Quantities that have the same value are represented by equal-sized bars, or units. These bar diagrams can be used to solve simple problems such as, "A toy car cost $5.70. A stuffed bear costs $3.80 more than the toy car. How much did the stuffed bear cost?" in level 2 and, "A watermelon is 5 times as heavy as a papaya. If the papaya weights 650g, find the weight of the watermelon," in level 3. It can also be used to solve more difficult problems, such as the somewhat infamous problem in level 4, "3000 exercise books are arranged in 3 piles. The first pile has 10 more books than the second pile. The number of books in the second pile is twice the number of books in the third pile. How many books are there in the third pile?". And a percentage problem in level 6, "Maryís salary is 10% more than Aliceís. If their total salary is $4200, what is Maryís salary?". These problems are all easy to solve by first diagramming them. The diagrams help the student pull all the information from the word problem into a pictorial representation and determine what equations to use. This also provides a pictorial introduction to the more abstract algebra since a bar of unknown length is similar to a variable.

The main feature of the Primary Mathematics series, the use of a concrete-to-pictorial-to-abstract process, runs throughout the whole series, both within specific concepts, and in an overall preparation for algebra. First, the concept is taught with concrete objects (from toy cars or other objects for addition and subtraction, to number discs for place-value or circles and bars for fractions, ratio, and percentage), then visually with pictures in the text and diagramming, and then abstractly with numerals and equations. A student, for example, will learn to find a missing part using objects, then relate that to a bar diagram, and then relate that to an equation such as 34 + ___ = 81.

Both flexibility and involvement are keys to using this program successfully. The textbooks are not laid out in neat, well-defined lessons, each with an opening activity, lesson plan, and closing activity. Some aspects of the curriculum are left mostly up to the teacher, such as how much and when to drill math facts. The textbook itself does not specify how to introduce the concepts concretely Ė that, too, is left up to the teacher. There are not lots and lots of practice problems for those students who need more practice. However, there are now guides which, though not scripted (the teacher should understand the material well enough to explain it), do provide background notes to help the teacher understand the topic better and how it fits in with the rest of the program, a suggested schedule, activities teaching the concepts, games and worksheets for working on math facts, and answers and solutions. There are now a number of supplementary books available which range from books with pages full of same-type problems to books that carry the concepts even deeper and beyond the Primary Mathematics texts and workbooks, with more-challenging word problems (5-step word problems instead of 3-step word problems, for example) and non-routine thinking-skill types of problems. So there is practically something for anyone who wants to take the time to fit the program to their childrenís needs. This is not an easy program to jump into midstream. A fifth-grade student cannot simply start with the 5A text, for example, when using Primary Mathematics for the first time, but it is an easy program to get "caught up" in by doing a few topics from earlier levels.

Primary Mathematics does teach things differently than U.S. texts do, so the homeschooling parentís involvement is important. A parent who has gone through the program once will have an easier time teaching it a second time ó seeing the program as a whole is extremely valuable to teaching a part of it effectively. Although some students are quite capable of using the later Primary Mathematics levels independently, and even the secondary levels (7th Ė 10th grade), the program was designed for interaction and discussion. If that is left out, some of the benefits are lost. Spending time with the program and being involved, rather than trying to get the student to do it independently, does have a good chance of renewing or beginning a parentís interest in math.

I did not get the Primary Mathematics because of international test scores, though it was nice to later discover that Singapore students scored highest on the TIMMS (The Trends in International Mathematics and Science Study which provides reliable and timely data on the mathematics and science achievement of U.S. students compared to that of students in other countries for an international perspective.] Neither was I concerned whether I was teaching the math just like they do in Singapore, any more than I was concerned whether I was teaching math like they do in schools here. I was just trying something new out of desperation, something that I hoped would get my kids past just going through the motions when solving math problems. Happily, I found that the Primary Mathematics and the secondary levels teach math like no other curriculum I tried. Even though my kids donít jump up and down for joy every time they "do" math, my 15-year old son does give it his highest acclaim, "Itís all right." J.H.