Mathematics and Education

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Comments on Milgram’s “Review of Final Draft Core Standards”

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Summary. In 2010 (apparently in June), Jim Milgram posted a review of the Common Core State Standards, comparing them with standards of high-achieving countries. In my opinion, his review misses some important details and makes some incorrect conclusions. 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 discuss some assertions in the review and give some detailed side-by-side comparisons of comments with standards, teacher’s guides, and other documents.

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

Referents: Textbooks and high-achieving countries

Examples in the review come from the 1998 California standards in Mathematics Framework for California Public Schools and (without citation) Singapore textbooks. The latter appear to be from the series first published in 1982, reprinted until at least 1999, and sold at http://www.singaporemath.com. The teacher’s manuals for this series are not mentioned.

The review briefly mentions the Russian textbook for grade 3 translated by the University of Chicago School Mathematics project (Pcholko et al., 1992). [Note: An earlier version of this post said “Russian textbook for grade 1 translated by the University of Chicago School Mathematics project (Moro et al., 1992).”]

The review mentions “high-achieving countries,” but does not indicate what those countries are. The table below lists some common referents for “high-achieving countries”: countries whose averages were well above the international averages for the grade 8 Trends in International Mathematics and Science Survey (TIMSS) or the Programme for International Student Assessment (PISA). Note that not all countries participated in each administration and that PISA is administered to 15-year-olds.

Test

TIMSS

PISA

Year

1995

2003

2007

2011

2003

2009

Belgium (Flemish)

Canada

Czech Republic

China, Macao

China, Shanghai

China, Taipei

Finland

Hong Kong SAR

Japan

Korea

Liechtenstein

Netherlands

Singapore

Switzerland

Sources: TIMSS results 2011, http://nces.ed.gov/timss/table11_3.asp; TIMSS results 2007, http://nces.ed.gov/timss/table07_1.asp; Highlights from PISA 2009, http://nces.ed.gov/pubsearch/pubsinfo.asp?pubid=2011004

Thus, Japan and Singapore predominate among high-achieving countries, as measured by high scores on TIMSS and PISA. As noted in the previous post, expectations in Japan and Singapore change slowly and are listed explicitly. Thus, details for expectations in Japan and Singapore can be illustrated by examples from pre-2010 textbooks and teacher’s manuals.

Algebra I in California: Is eighth grade Algebra I working?

. . . California standards where the expectation is that most students will be ready for Algebra I by eighth grade. Moreover, as the following graph shows, eighth grade Algebra I is basically working already, with almost 60% of California’s students taking the course either in seventh or eighth grade. (Milgram, p. 12)

Most countries do not have courses such as Algebra I and the relationship between tests of Algebra I proficiency and performance on other measures is not well studied. Moreover, the effect of standards (as opposed to other policy changes) is hard to determine. With those caveats, California performance on two recent measures is given below. Whether or not more California students are ready for Algebra I, their performance on these measures is, in general, below the U.S. average. Details are below. 

TIMSS 2011. California participated separately from the rest of the United States on TIMSS (see IES Newsletter, July 2010). For grade 8 mathematics, its average score was 493, which was below the US average of 509 as well as that of Singapore (611) and Korea (613).

National Assessment of Educational Progress. California was below the national average for grades 4 and 8 mathematics. For further analysis of California’s 2011 NAEP performance with respect to student demographics and as compared with Florida, Illinois, New York, and Texas, see pp. 16–19 of Mega-States: An Analysis of Student Performance in the Five Most Heavily Populated States in the Nation.

California and national public school performance on grade 8 NAEP

Source: http://nces.ed.gov/nationsreportcard/states/. Orange (as in the California average for 2011) indicates lower than national public. Tan indicates not significantly different from national public.

Computation in grades 1–4

In the section on “basic arithmetic and arithmetic operations” (pp. 1–3; pp. 14–15), the review refers to strategies mentioned in 1.OA.6 such as “counting on” or “making ten” as “unconstrained algorithms” (p. 3) and “special tricks for doing arithmetic” (p. 14). Learning these is seen as interfering with later learning of standard algorithms.

This neglects two things:

The CCSS distinguish between “strategies” and “algorithms.” In the CCSS, “counting on” or “making ten” are considered to be strategies rather than algorithms.

High-achieving countries such as Japan and Singapore teach the “special tricks” listed in standards such as 1.OA.6. However, this is not evident in their “standards” or textbooks, but elsewhere, in teacher’s manuals and articles on cross-national research, e.g., Fuson and Li (2009, pp. 796–799). Note that Fuson was a member of the CCSS feedback group. She has conducted cross-national research in mathematics education for over twenty years.

Details from the Singapore teacher’s manuals about the teaching of these strategies are given in the side by side comparison at the end of this post.

Common Core standards for grade 1: Behind, ahead, or unique?

Also among these difficulties are that a large number of the arithmetic and operations, as well as the place value standards are one, two or even more years behind the corresponding standards for many if not all the high achieving countries. (Milgram, p. 2)

The review does not give evidence for this claim. The side by side comparison at the end of this post gives counterexamples from some of the high-achieving countries listed above, mainly Singapore.

Other Common Core standards (e.g., 1.NBT.1 and 1.G.3) are criticized for being unusual or ahead of expectations in high-achieving countries. The table at the end of this post gives these criticisms with counterexamples.

Remarks 

Note that the comments above and examples in the side by side comparison below are not intended to be comprehensive. However, the following post discusses expectations in Japan and Singapore that are related to other aspects of Milgram’s review.

References 

References for standards and course of study documents used in this post are given in the previous post on national context. Note that curriculum documents for several countries (including China, Hong Kong, Japan, Singapore) may be downloaded here: http://hrd.apec.org/index.php/Mathematics_Standards_in_APEC_Economies.

California Department of Education. (1999). Mathematics framework for California public schools. Sacramento, CA: Author.

Curriculum Development Institute of Singapore. (1994). Primary mathematics 1A: Teacher’s guide (3rd ed.). Singapore: Federal Publications.

Curriculum Development Institute of Singapore. (1994). Primary mathematics 1B: Teacher’s guide (3rd ed.). Singapore: Federal Publications.

Hironaka, H. et al. (2006). Mathematics for elementary school 1 (M. Yoshida et al., Trans.). Tokyo: Tokyo Shoseki.

Fuson, K., & Li, Y. (2009). Cross-cultural issues in linguistic, visual-quantitative, and written-numeric supports for mathematical thinking. ZDM Mathematics Education, 41, 793–808.

Milgram, R. J. (n.d.). Review of final draft Core Standards, ftp://math.stanford.edu/pub/papers/milgram/final-report-for-validation-committee.pdf

Pchoiko, A. S., Bantova, M. A., & Moro, M. I. (1992). Russian grade 3 mathematics (9th Ed., R. H. Silverman, Trans.). Chicago: University of Chicago School Mathematics Project. (Original work published 1978) Available from the University of Chicago School Mathematics Project. The listing says “Mathematics textbook used in grade 3 (our grade 4) in Russia.”

Excerpts from Milgram’s review

Comments

Computation in grades 1–4  
Note that most of these standards [1.OA.6, 2.OA.2, 3.OA.7, 2.NBT.5, 3.NBT.2, 4.OA.4, 4.OA.6] have some sort of fluency requirement for operations in a range, but no requirement that the algorithm being used is either general or generalizable. (p. 2)   In this list, only 3.NBT.2 uses the word “algorithm.” Some of the other standards listed use the word “strategy.” As indicated in the CCSS glossary (p. 85), “strategy” is not used as a synonym for “algorithm.”

Computation algorithm. A set of predefined steps applicable to a class of problems that gives the correct result in every case when the steps are carried out correctly. See also: computation strategy.

Computation strategy. Purposeful manipulations that may be chosen for specific problems, may not have a fixed order, and may be aimed at converting one problem into another. See also: computation algorithm.

[In discussion of 4.OA.6] this is part of a definition that is given at least one year earlier in virtually all the high achieving countries at that.

Specifically, subtraction is defined in the following way: a − b is that number c, if it exists, so that b + c = a, while division is defined by a ÷ b is that number, c, if it exists, so that b × c = a. (p. 2)

[colors added to indicate connections with excerpts in right column and with 1.OA.6 below]

Use of this definition occurs in the California standards for addition and subtraction in grade 1:

2.2 Use the inverse relationship between addition and subtraction to solve problems. (p. 27)

In the CCSS, this definition is given in 1.OA.4. Its use occurs in 1.OA.6 as:

using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4). . . .

1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13). This standard misuses “Fluent,” I believe. One does not want students to develop fluent command of special tricks for doing arithmetic, as this could very well result in severe difficulties un-learning these methods in later grades where Core Standards asks for at least some degree of proficiency with standard algorithms. [colors added to indicate connections with excerpts in right column] These “special tricks” are part of the curriculum in Singapore. Some examples are below.These are some of the headings and objectives listed in the Singapore Primary Mathematics 1A, Teacher’s Guide for the first half of grade 1 (which has nine units).

Unit 2: Number bonds

To name the missing part in a number bond. [“Number bonds are given combinations of two numbers that make up a given number. Each number bond represents a part-whole relationship between three numbers. It is related to a family of four basic addition and subtraction facts. So the pupils may use the number bonds to help them add and subtract.” p. 15]

To make ten. (p. 13)

Unit 3: Addition

To use the “count on” strategy to add two numbers within 10, one of which is 1, 2 or 3.

To make ten. (p. 24)

Unit 6: Numbers within 20

To add two 1-digit numbers using the “make ten” strategy. (p. 71)

To add two numbers, one of which is 1, 2 or 3, using the “count on” strategy. (p. 72)

These are some of the headings and objectives listed in Primary Mathematics 1B, Teacher’s Guide for the second half of grade 1 (which has nine units):

Unit 3: Numbers to 40

To use the “count on” strategy to add two numbers within 40, one of which is 1, 2 or 3. (p. 15)

To add a 2-digit and 1-digit number with renaming [regrouping] using the “make tens” strategy.

To subtract a 1-digit number from a 2-digit number with renaming [regrouping] using the “subtract from 10” strategy. (p. 16)

Another mathematician (who was a member of the CCSS development team) views “making a ten” as an extremely important strategy. See section II of Roger Howe’s “Three Pillars of First Grade Mathematics.”

CCSS grade 1: Behind, ahead, or unique?
1.NBT.1. Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral. 120 is an extremely strange limit. What is the justification for stopping here? To my knowledge, there is no high-achieving country that has 120 as a limit. Also, the CA standard is counting to 100, as are the limits for many of the high achieving countries. (p. 15)  See pp. 91–92 of the Japanese first grade textbook Mathematics for Elementary School 1 for an example of a number line with labels from 0 to 120 and exercises that involve reading numerals up to 119 (with an illustration that asks “What number comes after 119?”).
Compare the first grade California Green dot standard

2.1 Know the addition facts (sums to 20) and the corresponding subtraction facts and commit them to memory.

with the much weaker standard in Core Standards:

1.OA.6. Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. . . . (p. 14)

Note the difference between “know” (California) vs “include” (Singapore) or “teachers can encourage” (Taiwan).Singapore grade 1 says under “mental calculation”:

Include:
• 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) [colors added to indicate connections with strategies in 1.OA.6.]

Japan: the first mention of mental calculation is grade 3 (p. 9).

 

1.G.3 Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares. This seems too early too me to be introducing fractional parts. Also, there is extreme vagueness in the notion of partitioning a circle into 4 equal parts. One often sees young students partition the circle as follows:  

[figure of circle partitioned by 3 equidistant line segments]

Students need a better feel for area than they are likely to have in first grade to handle this standard. (pp. 14–15)

Students can cut or fold paper to partition circles and rectangles as described for Unit 6: Halves and Quarters in the Singapore Primary Mathematics 1B, Teacher’s Guide. The three objectives of the unit are:

To fold a piece of paper into halves/quarters.

To recognise and name one half of a whole which is divided into 2 equal parts.

To recognise and name one quarter of a whole which is divided into 4 equal parts. (p. 59)

This corresponds to the section on halves and quarters in the Singapore student workbook for 1B, pp. 60–61.


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Written by CK

August 4, 2013 at 3:55 pm

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  1. […] make. Over at her blog Mathematics and Education she has just published two thoughtful critiques, one about Jim Milgram’s 2010 review and another about Jonathan Goodman’s comparison of […]


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