Tuesday, August 31, 2010

Specific Suggestions to Implement the Concerns Raised in “A Plea for Critical Revisions to the Common Core State Standards for Mathematics” Developed by Susan Jo Russell and Steve Leinwand as a follow-up to a conversation with Bill McCallum, P


Following up on “A Plea for Critical Revisions to the Common Core State Standards in Mathematics”, we are pleased to supplement our general concerns with the following grade by grade suggestions for improving the elementary grades section of the Public Discussion Draft of the standards.  Our concerns, as discussed with you on April 5, 2010, fall into two broad categories that are addressed in the two charts that provide our specific suggestions for revision and improvement:
1)    Our belief that the Number-Base Ten domain is rushed and that nothing is lost by shifting the grade placement of some of the content to provide more time for the development of algorithmic mastery; and
2)    Our belief that the Number-Fraction domain in grades 3-5 can be strengthened – kept challenging, but made more reasonable.


Part I.  Number – Operations and Number – Base Ten

Grade            Domain            Suggested Revisions

K            NOP            Move to Grade 1:           
5. Understand that addition and subtraction are related. For example, when a group of 9 is decomposed into a group of 6 and a group of 3, this means not only 9 = 6 + 3 but also 9 – 3 = 6 and 9 – 6 = 3.        
                                
K&! nbsp;         &nb! sp; NBT            Move to Grade 1:
1. Understand that 10 can be thought of as a bundle of ones—a unit called a “ten.”
2. Understand that a teen number is composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
3. Compose and decompose teen numbers into a ten and some ones, e.g., by using objects or drawings, and record the compositions and decompositions in base-ten notation. For example, 10 + 8 = 18 and 14 = 10 + 4.
4. Put in order numbers presented in base-ten notation from 1 to 20 (inclusive), and be able to explain the reasoning.
5. Understand that a de! cade word refers to one, two, three, four, five, six, seven, e! ight, or nine tens.
6. Understand that the two digits of a two-digit number represent amounts of tens and ones. In 29, for example, the 2represents two tens and the 9 represents nine ones.
                          &! nbsp;        
1             NOP            Insert from Grade K           
5. Understand that addition and subtract! ion are related. For example, when a group of 9 is decomposed into a group of 6 and a group of 3, this means not only 9 = 6 + 3 but also 9 – 3 = 6 and 9 – 6 = 3.
                                   
1            NBT            Insert from Grade K
1. Understand that 10 can be thought of as a bundle of ones—a unit called a “ten.”
2. Understand that a teen number is composed of a ten and on! e, two, three, four, five, six, seven, eight, or nine ones.
3. Compose and decompose teen numbers into a ten and some ones, e.g., by using objects or drawings, and record the compositions and decompositions in base-ten notation. For example, 10 + 8 = 18 and 14 = 10 + 4.
4. Put in order numbers presented in base-ten notation from 1 to 20 (inclusive), and be able to explain the reasoning.
5. Understand that a decade word refers to one, two, three, four, five, six, seven, eight, or nine tens.
6. Understand that the two digits of a two-digit number represent amounts of tens and ones. In 29, for example, the 2represents two tens and the 9 represents nine ones.

                                
1            NBT                        Move to Grade 2:           
7. Understand that in adding or subtracting two-digit numbers, one adds or subtracts like units (tens and tens, ones and ones) and sometimes it is necessary to compose or decompose a higher value unit.
8. Given a two-digit numbe! r, mentally find 10 more or 10 less than the number, without h! aving to count.
9. Add one-digit numbers to two-digit numbers, and add multiples of 10 to one-digit and two-digit numbers.
10. Explain addition of two-digit numbers using concrete models or drawings to show composition of a ten or a hundred.
11. Add two-digit numbers to two-digit numbers using strategies based on place value,! properties of operations, and/or the inverse relationship between addition and subtraction; explain the reasoning used.
                                   
2            NBT            Insert from Grade 1:           
7. Understand that in adding or subtracting two-digit numbers, one adds or subtracts like units (tens and tens, ones and ones) and sometimes it is necessary to compose or decompose a higher value unit.
8. Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count.
9. Add one-digit numbers to two-digit numbers, and add multiples of 10 to one-digit and two-digit numbers.
10. Explain addition of two-digit numbers using concrete models or drawings to show composition of a ten or a hundred.
11. Add two-digit numbers to two-digit numbers using strategies based on place value, properties of operations, and/or the inverse relationship between addition and subtraction; explain the reasoning used.
                                
2            NBT            Move to grade 3:           
13. Compute sums of two three-digit numbers, and compute sums of three or four two-digit numbers, using the standard algorithm; compute differences of two three-digit numbers using the standard algorithm.
                                
2            NBT                 &n! bsp;      Move to grade 5 as culminating idea about algorithms:           
10. Understand that algorithms are predefined steps that give the correct result in every case, while strategies are
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. For example, one might mentally compute 503 – 398 as follows! : 398 + 2 = 400, 400 +100 = 500, 500 + 3 = 503, so the answer ! is 2 + 1 00 + 3, or 105.
                                   
2          &! nbsp; NOP            New standard:        
Estimate sums and differences of two digit numbers using strategies based on rounding and place value; justify the estimates.
            !           &n! bsp;            
3            NBY            Insert from Grade 2:
13. Compute sums of two three-digit numbers, and compute sums of three or four two-digit numbers, using the standard algorithm; compute di! fferences of two three-digit numbers using the standard algorithm.

3            NBT        Revise #7:
Original #7. Understand that the distributive property is at the heart of strategies and algorithms for multiplication and division computations with numbers in base-ten notation; use the distributive property and other properties of operations to explain patterns in the multiplication table and to derive new multiplication and division equatio! ns from known ones. For example, the distributive property makes it possible to multiply 4 × 7 by decomposing 7 as 5 + 2 and using 4 × 7 = 4 × (5 + 2)= (4 × 5) + (4 × 2) = 20 + 8 = 28.        
REVISE (based on the fact that most of this appears, appropriately as 4-NBT 3a and 3b) as follows:
7. Understand that the distributive property is at the heart of strategies for multiplication computations with numbers in base-ten notation; use the distributive property and other properties of operations to explain patterns in the multiplication table and to derive new multiplication expressions from known ones. For example, the distributive property makes it possible to multiply 4 × 7 by decomposing 7 as 5 + 2 and using 4 × 7 = 4 × (5 + 2)= (4 × 5) + (4 × 2)! = 20 + 8 = 28.

3            NOP   Revise #7:           
Original #7. Solve word problems involving multiplication and division within 100, using an equation with a symbol for the unknown to represent the problem. This standard is limited to problems with whole-number quantities and whole-number quotients. Focus on situations described in the Glossary, T! able 2.        
REVISE (to  parallel 1 NOP-7) as follows: Solve word problems involving multiplication and division within 100, using objects, drawings and equations to represent the problem. This standard is limited to problems with whole-number quantities and whole-number quotients. Focus on situations described in the Glossary, Table 2.

3            NOP            New standard
Estimate sums and differences of numbers using strategies based on rounding and place value; justify the estimates.

3             NOP          Move to Grade 4           
9. Understand that multiplication and division can be used to compare quantities (see Glossary, Table 2); solve
multiplicative comparison problems with whole numbers (problems involving the notion of “times as much”).

4            NBT          Revise #6-8 to limit to 1 digit at grade 4, 2 digits at grade 5:           
ORIGINAL:
6. Compute products and whole number quotients of two-, three- or four-digit numbers and one-digit numbers, and compute products of two two-digit numbers, using strategies based on place value, the properties of operations, and/or the inverse relationship between multiplication and division; explain the reasoning used.
7. Explain why multiplication and division strategies and algorithms work, using place value and the properties of
operations. Include explanations suppo! rted by drawings, equations, or both. A range of reasonably ef! ficient algorithms may be covered, not only the standard algorithms.
8. Compute products of two-digit numbers using the standard algorithm, and check the result using estimation.
REVISED:
6. Compute products and whole number quotients of two-, three- or four-digit numbers and one-digit numbers, using strategies based on place value, the properties of operations, and/! or the inverse relationship between multiplication and division; explain the reasoning used.
7. Explain why multiplication strategies work, using place value and the properties of operations. Include explanations supported by drawings, equations, or both. A range of reasonably efficient algorithms may be covered, not only the standard algorithms.

4            NOP            New standard:           
8. Estimate products and quotients of two-, three- or four-digit numbers and one-digit numbers, using strategies based on rounding, compatible numbers and place value; justify the estimates.
!

4             NOP            Insert from grade 3:           
9. Understand that multiplication and division can be used to compare quantities (see Glossary, Table 2); solve
multiplicative comparison problems with whole numbers (problems involving the notion of “times as much”).

5            NBT            Insert, as revised, to focus on 2 digits, from Grade 4:           
6. Compute products of two two-digit numbers using strategies based on place value and the properties of operations; explain the reasoning used.
7. Explain why multiplication and division strategies and algorithms work, using place value and the properties of
operations. Include explanations supported by drawings, equations, or both. A range of reasonably efficient algorithms may be covered, not only the standard algorithms.
8. Compute products of two-digit numbers using! the standard algorithm, and check the result using estimation! .

5             NBT    Revised  to focus on one-digit divisors in grade 5 and two-digit divisors in grade 6:           
ORIGINAL:
1. Compute quotients of two-, three-, and four-digit whole numbers and two-digit whole numbers using strategies based on place value, the properties of operations, and/or the inverse relationship between multiplication and division; explain the reasoning used.
2. Explain why division strategies and algorithms work, using place value and the properties of operations. Include
explanations supported by drawings, equations, or both. ! A range of reasonably efficient algorithms may be covered, not! only th e
standard algorithm.
3. Use the standard algorithm to compute quotients of two-, three- and four-digit whole numbers and two-digit whole numbers, expressing the results as an equation (e.g., 145 = 11 × 13 + 2 or 120 ÷ 7 = 17 1/7).        
REVISED:
1. Compute quotients of two-, three-, and four-digit whole numbers and one-digit whole numbers using strategies based on place value, the properties of operations, and/or the inverse relationship between multiplication and division; explain the reasoning used.
2. Explain why division strategies work, using place value and the properties of operations. Include ex! planations supported by drawings, equations, or both. A range ! of reaso nably efficient algorithms may be covered, not only the standard algorithm.
3. Use the standard algorithm to compute quotients of two-, three- and four-digit whole numbers and one-digit whole numbers, expressing the results as an equation (e.g., 145 = 11 × 13 + 2 or 120 ÷ 7 = 17 1/7).

5            NBT            Insert from Grade 2:           
10. Understand that algorithms are predefined steps that give the correct result in every case, while strategies are
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. For example, one might mentally compute 503 – 398 as follows: 398 + 2 = 400, 400 +100 = 500, 500 + 3 = 503, so the answer is 2 + 100 + 3, or 105.


6            XXX            Insert from Grade 5:           
1. Compute quotients of two-, three-, and four-digit whole numbers and two-digit whole numbers using strategies based on place value, the properties of operations, and/or the inverse relationship between mul! tiplication and division; explain the reasoning used.
2. Explain why division strategies and algorithms work, using place value and the properties of operations. Include
explanations supported by drawings, equations, or both. A range of reasonably efficient algorithms may be covered, not only the
standard algorithm.
3. Use the standard algorithm to compute quotients of two-, three- and four-digit whole numbers and two-digit whole numbers, expressing the results as an equation (e.g., 145 = 11 × 13 + 2 or 120 ÷ 7 = 17 1/7).



Part II.  Number – Fractions

NOTE: for ease of reading, the revised standards only are listed in this domain; it will be useful for readers to have the CCSS draft nearby for reference when reading this section.

Grade            Domain            Suggested Revisions
3              NF               &nbs! p;     Fractional quantities and representations, suggested revisions:
           
1.  Understand that a unit fraction corresponds to a point on a number line and represents one part of a whole divided into equal parts.  For example, on a number line! , when t he interval from 0 to 1is divided into three equal parts, 1/3 is the endpoint of the first part.  If a whole is divided into three equal parts, one of the parts is 1/3 of the whole.  Students work with fractions with denominators of 2, 3, 4, 6, and 8.  (Revised old #1 to include both no. line and whole divided into parts and to specify fractions.  The two ideas could be listed as a) and b) to make them clearer.)

2.  Understand that fractions are built from unit fractions.  For example, ¾ represents 3 of 4 equal parts of a whole and 5/4 represents the point on a number line obtained by marking off five lengths of ¼ to the right of 0. Students work with fractions with denominators of 2, 3, 4, 6, and 8.  (Revised old #2 to include example with whole and to specify fractions.)

3.  Understand that two fractions are equivalent (represent the same number) when both fractions correspond to the same point on a number line or represent the same portion of a whole.  Recognize and generate equivalent fractions with denominators 2, 3, 4, 6, and 8 (e.g., ½ = 2/4, 4/6 = 2/3), and explain the reasoning using number lines, pictures of wholes, tape diagrams, area models, and story contexts. (Revised old #3 to include idea of whole, to specify fractions, and to reference representations.)

4.  Understand that whole numbers can be expressed as fractions.  For example, on a number line or using a representation of a whole, show that 1 = 4/4 and that 6 = 6/1. (Revised old #4—eliminated 7 = (4 x 7)/4.  This fits better in grade 4.)

(Old #5 is now not necessary, because it is included in #1.)

5.  Compare and order fractional quantities with equal numerators or equal denominators, using the fractions themselves, number line representations, pictures of a whole, tape diagrams, area models, and story contexts.  Use > and < symbols to record the results of comparisons. Students work with fractions with denominators of 2, 3, 4, 6, and 8.  (Revised old #6 to include consistent representations and to specify fractions.)

4            NF            Operations on fractions, suggested revisions:

1.  Compare and order fractional quantities, including those that have neither equal numerators nor equal denominators, using fraction benchmarks (e.g., greater or less than ½), number line representations, pictures of a whole, tape diagrams, area models, and story contexts.  Use =, >, and < to record the results of comparisons. Students work primarily with fractions with denominators 2, 3, 4, 5, 6, 8, 10, and 12. (New standard in order to add continued work on fraction equivalence, progressing from the work in grade 3; also included = as needed symbol.)

2.  Understand addition of fractions:
a.  Adding or subtracting fractions with the same denominator means adding or subtracting copies of unit fractions.  For example, 2/3 + 4/3 is 2 copies of 1/3 plus 4 copies of 1/3, or 6 copies of 1/3 in all, that is, 6/3.
b.  Sums of related fractions can be computed by replacing one with an equivalent fraction that has the same denominator as the other.  For example, the sum of the related fractions 2/3 and 1/6 can be computed by rewriting 2/3 as 4/6 and computing 4/6 + 1/6 = 5/6
 Students work primarily with fractions with denominators 2, 3, 4, 5, 6, 8, 10, and 12.  (Old #1 with specification of fractions added.)

3.  Add and subtract fractions with equal or related denominators within 1 (e.g., 1/8 + 5/8, ½ + ¼, 3/5 + 3/10, 7/8 – ¼), and solve word problems involving these operations.  Include explanations supported by drawings, equations, or both. Students work primarily with fractions with denominators 2, 3, 4, 5, 6, 8, 10, and 12.  (Revised old #2 to include representation, in order to be consistent with #4; specification of fractions added; tried to put the two objectives together so they seem related instead of just two different standards; also c! hanged “li
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