Mathematics and Education

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Comments on Goodman’s “Comparison of Proposed US Common Core Math to Standards of Selected Asian Countries”

with one comment

Summary. In July of 2010, Jonathan Goodman published a comparison of Common Core State Standards with curriculum documents from several Asian countries (China, Hong Kong, Japan, Singapore, Taiwan). In my opinion, his analysis has some serious flaws. In this post, I give some examples. In an earlier post, I have given a brief overview of differences in national context, noting the different uses of standards and other documents in the U.S. and elsewhere. These different contexts and uses suggest how a U.S. reader’s expectations may lead to misinterpretation of documents from outside the U.S. In this post, I compare some of Goodman’s statements with the content of these documents in two ways: comments and detailed side-by-side comparisons.

Caveat 1. A limitation of Goodman’s analysis is that he does not give concrete referents in many cases, either to URLs for documents or titles of documents. In general, referents can be guessed, from documents available at the web page Mathematics Standards in APEC Economies. However, in some cases, he gives page numbers that do not correspond to the content he describes. (Specific cases are noted below.) The documents that I discuss are in my list of references. A further limitation is that Goodman’s review has no page numbers, so I do not give page numbers in quoting from his review.

Caveat 2. The examples below are not meant to be comprehensive, but to suggest caution with respect to the review’s assertions and conclusions.


Singapore grades 7–10

Singapore has a 6-4-2 system: six elementary grades, four secondary grades, and two “preuniversity” grades. The first ten grades are part of compulsory schooling. The associated mathematics requirements described in two separate documents: Mathematics Syllabus Primary (for grades 1–6); Secondary Mathematics Syllabuses (for grades 7–10). Secondary students are offered different choices of content and pacing for mathematics. The four-year Express course taken by about 60% of students prepares them for the O level examination. The four-year Normal Academic (taken by about 25% of secondary students) or Normal Technical (taken about 15% of secondary students) are preparation for the N level examination. The Singapore Ministry of Education illustrates the general scheme of three multi-year courses followed by examinations as a graphic.

Another difference is that about 60% of secondary students take additional O or N level mathematics in grades 9 and 10 (Soh, 2008, pp. 26, 30). Thus, content for students in these grades varies. The syllabus groups content in upper secondary (corresponding to U.S. grades 9–10) in bands rather than giving it by grade level for “greater flexibility in teaching and learning” (Soh, 2008, p. 32). The upper secondary band is given by the notation 3/4 in the syllabus.

Goodman does not mention these differences in course options and seems to have assumed that Singapore “secondary” meant grades 6–10 rather than grades 7–10. Some examples consistent with this assumption are given in the table below together with excerpts from the Secondary Mathematics Syllabuses. The percentages in the second row are the percentages of students who take these courses as reported in 2008.

Excerpts from Goodman review

Topic location in Singapore syllabus







Grade 6 . . . Singapore graphs linear functions, connects this to the algebraic idea of proportionate change, and discusses the slope of a line.

Secondary one, i.e., grade 7 (p. 7). Secondary two, i.e., grade 8 (p. 19). Secondary two, i.e., grade 8 (p. 27).*
Grade 7 . . . Singapore has the Pythagorean theorem, probably without proof.

Pythagorean Theorem, secondary two, i.e., grade 8 (p. 11).

Pythagorean Theorem and trigonometry secondary 3/4, i.e., grade 9 or 10 (p. 23). Pythagorean Theorem, secondary two (p. 28); Pythagorean Theorem and trigonometry secondary 3/4, i.e., grade 9 or 10 (p. 31).
Grade 7. . . . and Singapore have a [sic] most of a full “Algebra I” class, with much practice in general algebraic manipulation, including rational expressions and solution of two linear equations in two unknowns. Appears to refer to “algebraic manipulation” and “functions and graphs,” secondary two i.e., grade 8 (p. 7). More rational expressions in secondary 3/4 (p. 9). Simple rational expressions and “functions and graphs,” secondary two (p. 19)
More complicated rational expressions, secondary 3/4 (p. 21)
Simple rational expressions and “functions and graphs,” secondary two (p. 27)
More complicated rational expressions, secondary 3/4 (p. 29)

*This illustrates an interesting feature of the Singapore syllabuses. Here the pace for the N(T) students (interested in a career in natural sciences or engineering) is slower than that for the students taking the Express course, perhaps allowing more depth.


In his comments about Singapore grades 6 and 7, Goodman is consistently off by a grade level. Moreover, he appears to have assumed that the Singapore syllabus’s “3/4” meant that the grades were combined, writing for grade 8, “I am dropping Singapore because they combine grades 8 and 9.”

Japan: Course of study vs teacher’s guide to course of study

As noted in the post on national context, Japan’s course of study documents are quite terse, listing topics to be included or excluded with little elaboration. Yet Goodman writes, “the Japanese standards (for example) give examples for almost every item to clarify precisely what the expectation is” and “I refer the reader to the Japanese standard, page 71, which has a beautiful, clear, and precise suggestion of how to teach column-wise addition.”

Goodman gives what appears to be a quote from Japan: “Transform algebraic expressions depending on purpose.” A Google search suggests that this sentence is unique to his review. However, a close approximation occurs on page 11 of the Junior High School Teaching Guide for the Japanese Course of Study: Mathematics (Grade 7–9): “Transforming algebraic expressions according to the purpose.”

This suggests that Goodman’s referent for “Japanese standards” is not (as is usually the case) the Japanese course of study, but rather the 2008 teacher’s guides for the course of study. These have different statuses. As stated in the beginning of the secondary guide translation:

While the Course of Study is regarded as a legal document, MEXT [Ministry of Education, Culture, Sports, Science and Technology] has published guides for the course of study, though they are not legally binding, to explain the gist of it. ‘The Teaching Guide of the Course of Study for Junior high School Mathematics’ is one of those guidebooks. (p. i)

Elementary grades

Calendar. Goodman writes of grade 1 expectations, “All but the US call for some understanding of the calendar. At least days, including the names of the days, and weeks.” I could not find instances of such expectations. (For details of what I did find, see the side by side comparison at the end of this post.)

Money and manipulatives. Goodman writes of grade 1:

All the other standards call for students to learn to manipulate the local currency. . . . The US standard calls for more abstract manipulatives rather than money. . . . The US standard puts money in grade 2, but with less emphasis on coins as manipulatives.

The statement about learning the local currency does not appear to hold for Japan (see details at the end of this post).

A more subtle issue is that expectations for learning about money such as Singapore’s “Include identifying coins and notes of different denomination” should probably not be interpreted advocating the use of coins as manipulatives outside of a special unit on money. Cross-national research on mathematics education and present-day curriculum materials suggest that money is probably not used as a manipulative in China, Japan, and Taiwan—and perhaps other Asian countries. For example, Harold Stevenson and James Stigler conducted studies of elementary mathematics education in the U.S., China, Japan, and Taiwan in the 1980s. They wrote in The Learning Gap:

Every elementary school student in Sendai possesses a “math set,” a box of colorful well-designed materials for teaching mathematics: tiles, clock, ruler, checkerboard, colored triangles, beads, and many other attractive objects. In Taipei, every classroom is equipped with a similar but larger set of such objects. In Beijing, where there is much less money available for purchasing such materials, teachers improvise. . . . There is also a subtle but important difference in the way Asian and American teachers use concrete objects. Japanese teachers, for example, use the items in the math set repeatedly throughout the elementary school years. (Stevenson & Stigler, 1992, p. 186)

See the web site of Global Education Resources for a present-day example of a Japanese “math set.”

Mental calculation. Goodman writes: “Most of the other countries call for students to be able to add and subtract “mentally” (without paper or manipulatives) up to about 20.” Countries such as Japan and Singapore do not include expectations for student performance in their course of study and syllabuses (see examples at the end of this post).


Note that documents for several countries (including China, Japan, and Singapore) may be downloaded here: Full references to these documents are listed in my earlier post.

Goodman, J. (2010, July 9). Comparison of proposed US Common Core math to standards of selected Asian Countries. Education News,

Ministry of Education, Culture, Sports, Science and Technology. (2010). Junior high school teaching guide for the Japanese Course of Study: Mathematics (Grade 7–9) (Masami Isoda, trans.). CRICED, University of Tsukuba. Available at

Soh, C. K. (2008). An overview of mathematics education in Singapore. In Z. Usiskin & E. Willmore (Eds.), Mathematics curriculum in Pacific Rim countries—China, Japan, Korea, and Singapore (pp. 23–36). Charlotte, NC: Information Age Publishing.

Stevenson, H., & Stigler, J. (1992). The learning gap. New York: Simon & Schuster.

Takahashi, A., Watanabe, T. & Yoshida, M. (2008). English translation of the Japanese mathematics curricula in the course of study, grades 1–9. Available at [translation of 2008 course of study]

Side-by-side comparisons for grade 1


Excerpts from Goodman’s review


For example, the US standard for grade 1 arithmetic (page 16) includes: “Add within 100, including adding a two-digit number and (sic) a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. . . .” In light of the math wars, it is important to state whether strategies means an exercise in student problem solving or practice with some form of a traditional algorithm. An example would clarify this.[colors added to indicate connections with comments and excerpts in right column, rows 1 and 2.] This comment seems to overlook the entries for computation algorithm and computation strategy on p. 85 of the CCSS glossary:
Grade 1. The six standards [i.e., standards of the six countries] have much in common. A major goal is addition and subtraction within a limited range: 0 to 20 for the US, and 0 to 100 for the others. This seems to omit CCSS’s 1.NBT.4 (shown in the row above), 1.NBT.5, and 1.NBT.6. In particular, 1.NBT.4 begins “Add within 100.” See 12 of the Singapore syllabus for 2007 gives similar expectations:

  • addition and subtraction within 100 involving

∗  a 2-digit number and ones,

∗  a 2-digit number and tens,

∗  two 2-digit numbers.

Page 14 of Hong Kong gives as part of an exemplar unit for the second half of grade 1:

Addition and subtraction (I)
(addition within 2 places; subtraction within 2 places, excluding decomposition)

Calendar. All but the US call for some understanding of the calendar. At least days, including the names of the days, and weeks. I can’t find expectations about names of days in the documents that I looked at. The closest that I’ve gotten is expectations about time. For example:

(iii) Acquire knowledge of year, month and day and familiarize with their interrelationships. (China, grades 1–3, p. 19)

Understand the relation between one hour, one day, one week, one month and one year. (Korea, grade 2, p. 14)

Students will understand elapsed time and be able to use it appropriately.
a. To become aware of days, hours, and minutes, and to understand the relationships among them. (Japan, grade 2, p. 5)

Money. All the other standards call for students to learn to manipulate the local currency. Some only use coins, in view of the overall restriction to numbers less than 100. Many call for students to practice place value and arithmetic with money, for example by exchanging pennies for nickels and dimes. Many call for students to read price tags and count out payments. The US standard calls for more abstract manipulatives rather than money. My opinion is that money is more natural. The US standard puts money in grade 2, but with less emphasis on coins as manipulatives. I have an electronic copy of the Japanese course of study. I have searched on “money,” “currency,” “yen” with no results. I also have copies of English translations of mathematics textbooks for elementary grades (Tokyo Shoseki, 2006). I can’t find money mentioned in the table of contents of any textbook for grades 1–6.The Singapore 2007 syllabus does include money in grades 1, 2, 3, 4, 5, 6. The teacher’s guide for primary 1B does mention changing a ten-cent coin to five-cent coins (p. 100) in a separate unit on money.

The Chinese 2001 curriculum standards for grades 1–3 say: “(i) Acquire knowledge of dollar, ten cents and cent in concrete situations and contexts, as well as understand their interrelationships.” I see no mention of using coins as manipulatives. Perhaps “concrete situations” was interpreted as involving manipulatives.

Mental arithmetic. Most of the other countries call for students to be able to add and subtract “mentally” (without paper or manipulatives) up to about 20. . . . Note the difference between “be able” vs “include” (Singapore) or “teachers can encourage” (Taiwan).Singapore grade 1 says under “mental calculation”:

• addition and subtraction within 20. (p. 12)

Taiwan grade 1 (emphasis added):

  • Teachers can encourage students to use mental calculations in proper basic adding and subtracting conditions and questions. But this does not mean the adding and subtracting questions are limited to basic addition and subtraction. For example: Students still use counting or decomposition and synthesis strategy to calculate the questions such as 12 + 9. (p. 102)

Japan, the first mention of mental calculation is grade 3:

As for the content A-(2), (3) students should be able to calculate simple calculations mentally. (p. 9)

A-(2), (3) are:

(2) Students will be able to add and subtract accurately and reliably and will further enhance their ability to use those operations appropriately.
a. To explore ways to add and subtract 3- and 4-digit numbers and to understand that those calculations are based on addition and subtraction of 1- and 2-digit numbers; to understand how to add and subtract using algorithms.
b. To add and subtract accurately and reliably, and to use those operations in different situations appropriately.
c. To explore properties of addition and subtraction and use them when considering ways of adding, subtracting, or checking answers.

(3) Students will extend their understanding of multiplication. They will be able to multiply accurately and reliably and use the operation in different situations appropriately.
a. To understand that multiplication of 2- and 3-digit numbers by 1- or 2-digit numbers is based on basic multiplication facts by exploring ways to complete the calculations; to understand how to multiply using an algorithm.
b. To be able to multiply accurately and reliably, and to be able to use multiplication appropriately.
c. To explore properties of multiplication and use them when considering ways of multiplying or checking answers.



Written by CK

August 4, 2013 at 4:12 pm

One Response

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  1. […] has just published two thoughtful critiques, one about Jim Milgram’s 2010 review and another about Jonathan Goodman’s comparison of CCSSM and the standards of certain Asian countries. […]

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