Professor of Mathematics
Courant Institute of Mathematical Sciences
New York University
July 9, 2010
Executive summary. The proposed Common Core standard is similar in earlier grades but has significantly lower expectations with respect to algebra and geometry than the published standards of other countries I examined. The Common Core standards document is prepared with less care and is less useful to teachers and math ed administrators than the other standards I examined. I have reservations about the Common Core standards regarding statistics in grades 7 and 8.
History and overview. At the request of Bill Evers and Zeev Wurman, I examined the published math standards of (South) Korea, Singapore, Hong Kong, and Japan, and Taiwan, as well as the proposed Common Core US standard, which I simply call US below. Evers and Wurman expressed particular interest in the question of teaching of Algebra in grade 8. They told me that this might be a major difference between the proposed Common Core national standard and the California State math standard. I have not examined the California standard.
The foreign countries chosen were from a recent list of high performing countries. Other high performing countries whose standards I have not examined are Australia and Flemish Belgium (Walloon Belgium has separate standards and is not in the top group by performance). Neither of these standards was available to me. The Flemish standard is not translated into English.
The conclusions and opinions expressed here are mine exclusively. Evers and Wurman have their opinions, they emphasized that they are not mathematicians and did not try to influence me, which would have been futile. I agreed to do the comparison because I have a strong interest in math education. They did provide me with translations of the foreign standards.
In this report, I offer judgments and opinions. For example I judge the arithmetic standards for first grade to be similar across countries. I try to label opinions as such. For example, it is my opinion that the Common Core draft description of the “mathematically proficient student” is so unrealistic as to be detrimental.
My experience with mathematics comes foremost from my career of mathematics research and teaching in a department ranked among the top ten nationally (US News). I also have considerable interest in K-12 math education, and have been actively involved in mostly fruitless attempts to influence New York City Department of Education math curriculum. I am familiar with the NCTM standards and ongoing debates on the best ways to teach math to kids.
Educational philosophy. All six of the standards documents I examined offer statements of educational philosophy. These statements have important similarities, including the need for understanding as opposed to rote, the need for research based age appropriate curricula, and the importance of higher level skills such as judgment, strategy, resourcefulness, and the ability to communicate. All of them state that high level skills do not come without learning facts and practicing algorithms.
I go through the standards grade by grade with comparisons, opinions, and some recommendations. I refrain from discussing many things the six standards have in common or that I do not think are significant differences. For example, all six standards call for first graders to “know their shapes”, but the details differ from country to country. I also mention what, in my opinion, are some strengths of non US standards documents that the US would do well to adopt.
In my opinion, the US standards are the least informative of the six. This will make it harder for US teachers and administrators to determine whether a particular curriculum or test meets the standard. By contrast, the Japanese standards (for example) give examples for almost every item to clarify precisely what the expectation is. The US standards are a study in bureaucratic ambiguity. 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 andstrategies 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.
Grade 1. The six standards 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. All but Korea include carrying and borrowing. All mention place value and abstract properties of addition and subtraction.
The US and most others include pre-algebra problems such as 4 + __ = 9. The US standard emphasizes students’ ability to explain in words the reasoning in the solution of a math problem or calculation.Â An example of a correct explanation would be most helpful. The Japanese standard includes working with a number line, which I think is a great idea (in grade 2 for the US). There are expectations in the other standards that are not included (at least not explicitly) in the US standard.
– 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.
– 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. The US standard does not. Singapore calls for memorization up to 9+9. The US standard puts this in grade 2.
– Calendar. All but the US call for some understanding of the calendar. At least days, including the names of the days, and weeks. Some call or understanding the yearly calendar, including months. In my opinion, there is value in this, but one could argue that memorizing the names of the days is not math.
Grade 2. The six standards have much in common. Almost every major topic mentioned on one is mentioned in all. The biggest exception is that Hong Kong has nothing about fractions. Multiplication is receives less emphasis in the US standard than in the other five. It is not clear whether the US standard calls for the operation of multiplication, and the multiplication symbol, to be defined.
The US standard for addition is ambiguous. Quoting from page 17 (italics mine): “…(students) develop, discuss, and use different, accurate, and generalizable methods to compute sums and differences of whole numbers in base-ten notation, using their understanding of place value and the properties of operations.” Page 19 has: “5. Fluently add and within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. 6. Add up to four two-digit numbers using strategies based on place value and properties of operations.” A standard should be clear on this point  but is not. Are second graders responsible for the column-wise addition algorithm or not? The wordings have much in common with the 1989 NCTM standards that deprecated algorithms. Will students develop the methods, or will teachers develop (i.e. teach) them? The terms accurate and generalizable also are from the math wars and (in my opinion) are out of place here. 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.
Grade 3. Again the six standards are similar. The most significant differences are as follows. The US emphasizes area more than the others. Japan, Hong Kong, and Korea omit area in grade 3. The US catches up with the other five in calling for addition algorithms.  Japan, Korea, and Taiwan include decimals as fractions n.x = n+(x/10). Japan includes using an abacus and the Japanese names for the powers of ten up to 10,000. Taiwan has “vertical” multiplication of a pair of two digit numbers. The other five have more on angles than the US.
Grade 4. The six standards handle fractions and decimals similarly. Some operations are applied to numbers of arbitrary size, with warnings not to overwhelm the kids with huge numbers. Highlighting the main differences: The US standard for adding and subtracting (the standard algorithm) and multiplying (multi-digit times one digit) whole numbers is what the other countries call for in grade 3. The other five give more emphasis to approximation and estimation than does the US. They also mandate areas, as the US did in grade 3. The difference (in treating areas) is that the US explicitly says to ignore units, while the other five call for explaining units of area, such as square meters. In my opinion, the US is wrong to ignore area units. It means treating area as purely geometric (a property of plane figures) rather than physical (a property of things like rugs, roofs, etc.). The US and Japan are better than the others (in my opinion) in calling for placing fractions and decimals on a number line. Korea and Hong Kong call for calculators to check arithmetic.
The contrast in the styles of the standards documents becomes more pronounced in grade 4. The US document treats some topics, such as fractions, in pedagogical detail. Others are left obscure, such as (page 29): “…multiply two two-digit numbers, using strategies based on place value and properties …”. By contrast, Taiwan has a whole page (page 139) devoted to illuminating examples of what is and is not mandated for fractions in grade 4. The US call for the standard algorithm makes it less clear, not more, what the methods are intended in grades 2 and 3.
Grade 5. The six standards are similar in how they cover arithmetic with fractions and decimals. The US is unique in introducing plane Cartesian coordinates. The US also is uniquely weak in several areas: the greatest common divisor/least common multiple, areas on non-rectangular regions such as triangles, ratios of quantities with units (e.g. miles per gallon, meters per second, kilograms per liter), percentages, and graphical representation of data (pie charts, bar graphs, etc.). Other countries are doing more to prepare students for algebra. Hong Kong has what I consider a great topic, understanding large numbers such as 10,000,000.
There is a fundamental difference in the treatment of geometry in grade 5. The other five discuss interesting (my opinion) topics like tilings, similarities, and polygons. The US standard is what I consider rote memorization of properties of figures. You can appreciate the difference by thinking how you would test the material. One might be: “Give a way to divide a 1 X 3 rectangle into a collection of four triangles?” The other might be: “In what way is a square different from a rectangle?”
The US standards document continues to puzzle me. After paying little attention to estimation in grades 3 and 4 (while the other five standards constantly emphasize it), now estimation returns. The description of Cartesian coordinates seems to imply that people might be unfamiliar with the concept: “Use a pair of perpendicular number lines, called axes, to define a coordinate system …” Cartesian coordinates are great (opinion), but they should be described in a way that will make sense to fifth graders. The document suggests measuring volumes by counting cubes, including cubic cm, cubic ft (will classrooms have piles of foot sized cubes?), and improvised units (cubic cubits?).
Grade 6. All six standards finish arithmetic, including multiplication and division of fractions and decimals, factoring of integers, and LCM/GCD for manipulating fractions. All discuss ratios, percentages, constant speed, and general proportionality expressed symbolically (e.g. d = 5t, where d is distance and t is time). All standards advance algebra using variables and abstract manipulation, but to different degrees. But the standards are much more different in grade six than for earlier grades. I take these differences by topic.
In pre-Cartesian (non-coordinate) geometry, the US catches up by doing areas of triangles, while the others pull ahead by doing areas and circumferences involving p.  Japan discusses making scaled copies of figures.
Only the US and Singapore have coordinate geometry. The US calls for determining distance between points whose x or y coordinates are identical (not very interesting in my opinion). Singapore graphs linear functions, connects this to the algebraic idea of proportionate change, and discusses the slope of a line (more interesting).
Only the US discusses negative numbers. Other countries do negatives in grade 7. Much time is devoted to arithmetic with whole numbers and fractions of arbitrary sign, distances between signed quantities, absolute value, and the “rational number line” (see pet peeves below). In the examples of naturally occurring negative numbers (negative temperatures, owing money) there is “negative electrical charge”. I highly doubt (my opinion) that any sixth grader will be able to appreciate the difference between positive and negative charge.
The US standard has much more on statistics than the others, including multiple measures of center (mean and median) and variation (quantile differences, mean absolute value deviation). Korea has some probability, but that part of their standard is vague. Japan and Hong Kong have mean, but not median or measures of variation. Taiwan, Singapore, and Korea stop at data representation without doing summary statistics. The US has less practice in graphical data presentation than the other five.
The other five countries all have less material in sixth grade than they had in fifth grade or the US has in sixth. I can only guess that this is considered a consolidation year for them, with much time spent honing skills.
The US standard for sixth grade is (in my opinion) careless and vague.
– (page 39): “…that a data distribution may not have a definite center and that different ways to measure center yield different values.” The italicized part is incorrect whatever center means, a data set has one by that definition. If it were deleted, the rest would be simpler and more correct.
– The standard calls for formulas involving powers, but powers (squares, cubes, etc.) have not been discussed yet. Is this their first introduction?
– There are many references to “real world examples” but no examples. I wonder what they have in mind for real world examples of distances between points in the plane with the same x coordinate.
– (page 45): “Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.” As a mathematician conducting research in computational statistics, I wonder what this could refer to. Maybe it refers to questions of robustness in which fat tailed distributions call for using the median rather than the mean. If it is a standard, people should know what it means.
Grade 7. I leave out Hong Kong because combines grades 7-9. The standards from Japan for grades 7 and 8 are less complete than for earlier grades. All five countries now use coordinate geometry at least to place points and graph lines. All have some algebraic manipulation, though the US has the least. All have a more thorough discussion of rates and proportionality with a variety of practical examples and graphical implications. All expect a full mastery of the rational number system (ratios of signed integers) that the US specified in grade 6 (and repeats in grade 7). The US catches up in geometry, including p, for example. Beyond this, the US differs from the others.
Korea has the least algebra, somewhat more than the US, particularly functional notation and exponents. It has much more geometry than the US, both in breadth (lots about polyhedra) and depth (systematic reasoning bordering on Euclid style theorem proving). It is unique in covering binary representations of numbers.
Taiwan and Singapore have a 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. Taiwan discusses exponents, particularly for representing powers of ten and scientific notation, which I strongly endorse (an opinion). Singapore has the Pythagorean theorem, probably without proof. Singapore also has basic probability and mean and median of a data set, but without measures of variability. Taiwan has no statistics or probability this year.
The US standard is distinguished from the others by its breadth and depth in statistics and basic probability. I have reservations about many of the specifics. The US standards document continues to be problematic. The grade 7 standard contains many helpful examples, which makes the many remaining vague items without examples less explicable. Here are some sample quibbles:
– I recommend removing the hydrogen atom as an example of charge cancellation. The background knowledge for this is missing: electrons, protons, atoms, charge, electric attraction (holding the atom together), …
– (page 50) suggests drawing “with technology”. Does this mean a computer drawing program that students would have to learn to use? If so, it would be a very large investment of student time for not very much educational gain. This kind of thing should not be ambiguous in a standards document.
– (page 50) states that someone (teacher, student?) should “give an informal derivation of the relationship between the circumference and area of a circle.” I imagine this might have to do with the area difference between circles of slightly different radius. Is that right?
– (page 51) Students are to choose words at random from a book. How? They are to use random digits to simulate a random process. Where do the digits come from?
– Most seriously, the standard calls for drawing “informal comparative inferences about two populations”. This amounts to testing the hypothesis of equality among sample means. The method suggested, comparing the difference between the means to twice the mean absolute deviation, has no basis in statistics. Any real test would depend on the sample sizes. This is an admirable topic, but the contents proposed are disinformation (a judgment).
– The notion of “probability sample space” is, in my opinion, not very helpful in this very elementary setting. I prefer the approach of Taiwan — discussing events as sets without taking time to define the whole space.
Grade 8. I am dropping Singapore because they combine grades 8 and 9. I compare the remaining standards (US, Korea, Taiwan, Japan) area by area.
Roots and irrational numbers: All have square roots, with the possible exception of Japan. Taiwan has more general rational exponents, but lightly. The US asks students to “know that the square root of 2 is not rational”. Does this mean being able to repeat that sentence or understanding the proof? The US standard asks students to tell that the square root of 2 is between 1.4 and 1.5 by truncating its decimal expansion. But where does the decimal expansion come from, a calculator?
Algebra: Both Korea and Taiwan call for generic operations of algebra such as factoring, multiplying polynomials, polynomial division (not in all cases), completing the square, etc. The US standard by contrast, supports a tightly circumscribed list of algebraic tasks centered on pairs of linear equations in two unknowns. The US catches up with exponents. It includes the mathematicians’ definition of function (set of ordered pairs so that …) but with an uncertain range of application. Included are general linear functions, at least one quadratic (with a proof that quadratics are not linear, as though a glance at the graph would not suffice), and possibly others that can be increasing and decreasing over different intervals of their argument (as functions are in calculus).
Geometry: Korea, Taiwan, and Japan call for significant work in the direction of traditional high school Euclidean geometric proof, but not as much as US high school geometry from the 1960s. This includes ruler and compass constructions (perpendiculars, bisections, congruences) and some of the easier proofs. The US calls for students to “understand” congruence of plane figures, but it is unclear what this means beyond the definition. US students are asked to know the Pythagorean theorem and its proof, but I question the wisdom of asking students to memorize something they do not have the background to appreciate. The US makes more of the geometric interpretation of pairs of linear equations than the other countries to.
Statistics: Only the US discusses statistical modeling and hypothesis testing. In grade 8, it discusses linear regression models and hypothesis testing on categorical data. But students have no systematic way to do either task. Linear fits are to be made “informally” from scatterplots. Goodness of fit is to be judged by eye. Categorical data decisions are made on (as far as I can tell) no basis whatsoever. If 60% of boys and 70% of girls pass math, is that a significant difference? How would one decide? In my opinion, this is negative education — giving kids the incorrect idea that they know something about regression and categorical data analysis. There is an AP statistics class in many high schools (both my kids took it). This has, for example, the ideas behind hypothesis testing, the role of statistical models, and the central limit theorem (informally). Making statistical inferences with less than this is dangerous.
Algebra. There are two styles of algebra, with the US on one side and most of the other countries on the other. Japan is in the middle. All ask students to understand abstract variables, such as x and y, and to know how to solve pairs of linear equations. All ask students to have seen manipulations such as dividing both sides of an equation by the same number to preserve the equality. The US (up to grade basically stops here. Japan includes multiplication and division of polynomials. The other countries go further to for example completing the square (solving general quadratics), manipulating rational functions, etc.
An item from the Japanese standard captures the difference: “Transform algebraic expressions depending on purpose.” General algebraic manipulation requires the student constantly to recall the mathematical principles and to plan strategy. Suppose, for example, a students wants to isolate a variable on one side of the equation. He or she must develop a strategy consisting of a sequence of operations. The strategy will be different from problem to problem. If manipulations are limited to pairs of linear equations, the student will simply memorize the operations required to solve them.
–Problem solving, rote, and patterns. Most people believe that math education should involve some serious independent creative problem solving. It also is a sad fact that students want to be told how to do things, and teachers like to tell them. Items introduced into the curriculum as problem solving can evolve into rote. This is nowhere more clear than in looking for patterns in sequences of numbers. The sequences offered tend to be arithmetic, and students are trained to look for common differences. The grade 4 patterns of the six curricula are all arithmetic progressions. But students could be presented with a wider range of patterns. For example, 1,2,1,2,1,2,… (easy but not arithmetic), 1,2,1,1,2,1,1,1,2,1,1,1,2,… (slightly harder), 1,1,2,3,5,8,13,23,… (a real challenge, but probably some kids would get it).
–Mathematical exactitude. Mathematicians have a bad reputation in the K-12 math ed world partly because of the “new math” disaster we helped create in the 1960s. My sixth grade math book from that period (found much later in my parents’ closet) had a discussion of sets that included (approximately) the sentence: “For any set and any property, you can form the subset of all members of the set that satisfy the property.” This probably was followed by an example such as picking out the apples from a set of pieces of fruit. As a recent math PhD, I recognized the aussonderung axiom of Zermelo Fraenkel set theory. I knew that this axiom has the purpose of ruling out Russell’s paradox. But how could it have helped a sixth grader? Mathematicians (in K-12 math ed discussions) must get used to things that are well enough understood even if not absolutely precise. We should avoid concepts that require more maturity than kids of a certain age have. We need not describe the coordinate plane in detail for fifth graders, or expect sixth graders to appreciate the distinction between a finite and infinite set of solutions. At the same time, we can prevent flatly incorrect statements such as (US, page 45): “Recognize a statistical question as one that anticipates variability in the data related to the question …”, which confuses a statistician’s model of random data variation with the fact of a single unchanging data set. Also (Taiwan, page 175): “For example, even though both 14 and 16 are composite numbers, they are coprime.” (Should have been 14 and 15?)
 The US is unique in having a standard for Kindergarten. I have combined the US kindergarten and grade 1 standards when comparing to grade 1 standards from the other countries.
 This opinion is stated as a judgment because I expect few to disagree.
 I do not know whether the plural is significant. The Everyday Mathematics curriculum adopted by the New York City Department of Education includes several variants of each of the algorithms of arithmetic. In my opinion, teaching multiple algorithms is a confusing waste of the students’ time.
 Hong Kong has a charming learning objective (page 44): “Tell the stories of ancient Chinese mathematicians discovering p.” I think this is time well spent.