Questions and answers from CCSS author, Bill McCallum

On pedagogy: Bill McCallumsays: April 13, 2012 at 10:19 am
Dear Lisa, I don’t see the standards as dictating any particular teaching method, but rather setting goals for student understanding. Different people have different ideas about what is the best method for achieving that understanding. That said, I think it’s pretty clear that classrooms implementing the standards should have some way of fostering understanding and reasoning, and classrooms where students are just sitting and listening are unlikely to achieve that.

On time and money:
Primary teachers become very emotional about the placement of the time and money standards. Kindergarten- no standards on time or money; Grade 1-telling and writing time, no money standards; Grade 2-time and money; Grade 3-no money standards. I have shared my view and would like to share your response which I am certain has more credibitlity.

April 13, 2012 at 10:32 am
Reading time and knowing the value of coins are important life skills, which students could learn in many places: in the home, in social studies, in science, in mathematics, in history, or in english language arts. There has been a tendency to overload mathematics standards in particular with these life skills, at the expense of more important work on number and operations. Perhaps this was because mathematics standards came along first, so putting these things there was a way of ensuring they were taught. The view of the Common Core is that, used in the right way, they can be tools for learning about number and operations, but they are not mathematics topics in their own right. If kids come to school with knowledge about them, or if there is a way of weaving them into the curriculum that supports the main focus, then that’s fine. But too often they become the main focus themselves. The strongest message of the Common Core is: focus on what’s important and give it the time it needs, so that kids have a chance to learn it well and progress onto other things. That required paring down previous standards.

On "Special Cases":
Cobb says: April 17, 2012 at 9:42 am
Thanks so much for taking time to repond to all of these questions. What it meant when in third grade NF 3 special cases?3NF3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

April 17, 2012 at 12:30 pm
Not sure what “special cases” may be, but the denominators in Grade 3 are limited to 2, 3, 4, 6, and 8. Although the CCSS does not limit the fractions to be discussed in Grade 3 to proper fractions, there are still infinitely many fractions with those denominators. However, I would suspect that the limit of equivalent fractions in Grade 3 will be basically the proper fraction situations, that is, 1/2 = 2/4 = 3/6 = 4/8, 1/3 = 2/6, 2/3 = 4/6, 1/4 = 2/8 and 3/4 = 6/8. Reply

April 17, 2012 at 12:54 pm
Tad has it right. The idea is to limit to situations where you can see the equivalence by direct reasoning from the definition of a fraction, but not get into the general way of seeing equivalence. For example, you might see that 1/2 is equivalent to 3/6 using a tape diagram divided into 2 and then into 6. But you wouldn’t get into 3/6 = 3×1/3×2 = 1/2.

On Fluency:
Leandra says: April 22, 2012 at 1:51 pm
Dr. McCallum,
I need help with clarifying the fluency with addition and subtraction facts in K-2.
K- Fluent with addition & subtraction w/in 51- Fluent with addition & subtraction w/in 102- Fluent with addition & subtraction w/in 20 AND knows from memory single-digit to 9+9 (add only)
We are working on standards based report cards for our 1-2 grade levels. We have standards based in Kinder already. This year in kinder, we listed the standard as shown and then, for assessment purposes only, we used flash cards to assess fluency. (We were sure to use number talks, images, and manipulatives to ensure understanding).
When we started working on standards based for 1 and 2 we ran into some confusion, because we had at first thought in K and 1 we should be “flash card fluent” under 5 and under 10 respectively, but then in 2nd it says know from memory for the ones they should be “flash card fluent” with and the word fluent has a slightly different meaning.
What wording could be used on a report card to differentiate these skills for parents? And if the K and 1 should not be “know from memory” how should teachers assess the facts in kinder and first? Reply

April 24, 2012 at 4:42 am
In November 2000 issue of Teaching Children Mathematics, Susan Jo Russell discussed what NCTM meant by “fluency.” She writes (p. 154):—Fluency, as used in Principles and Standards, includes three ideas: efficiency, accuracy, and flexibility.• Efficiency implies that the student does not get bogged down in many steps or lose track of the logic of the strategy. An efficient strategy is one that the student can carry out easily, keeping track of subproblems and making use of inter- mediate results to solve the problem.• Accuracy depends on several aspects of the problem-solving process, among them, careful recording, the knowledge of basic number com- binations and other important number relation- ships, and concern for double-checking results.• Flexibility requires the knowledge of more than one approach to solving a particular kind of problem. Students need to be flexible to be able to choose an appropriate strategy for the prob- lem at hand and also to use one method to solve a problem and another method to double-check the results.—I don’t know if this view is consistent with the CCSS writers’ view, but I like (yes, my personal preference) this view on fluency. It should also be noted that “fluency” seen from this perspective does not necessarily mean “quick.”

I also think it is interesting that the CCSS distinguish “fluency within 20″ and “knows from memory ” up to 9+9. This seems to suggest that students should be fluent with calculations like 13+5 and 18-3 using their understanding of the meaning of operations and number sense. But, for 1+1 … 9+9, the CCSS seems to expect students to “just know” the facts. I also like the fact that the CCSS puts “memorization” AFTER fluency. I think if students become fluent (as explained by Russell), they will remember basic facts, too.

April 26, 2012 at 8:00 am
Thanks Tad. So in your opinion, would you start assesessing “know from memory” addition facts only in kinder under 5 and under 10 in 1st as a progression toward the 2nd grade standard?

Or would you just expect them to be able to do them “unhaltingly” but not necessarily from memory?

For our second graders we are using flash cards to assess during individual student interviews and we expect them to know the fact within 3 seconds. Thoughts?

April 26, 2012 at 1:52 pm
First, “fluently” refers to how you do a calculation, whereas “know from memory” means being able to produce the answer when prompted without having to do a calculation. In CCSS, “fluent” means “fast and accurate.” The sort of flexibility that Tad is talking about is coded into many of the standards that are not explicitly about fluency, so it is part of the standards as a whole. I note that Tad says “fluent” does not necessarily imply “quick”, whereas I have said that it does imply “fast”. So there seems to be a bit of disagreement there, although maybe not that much; “fast” for a Kindergartner is not as fast as “fast” for a 2nd grader. If a Kindergartner adds numbers within 5 by saying the starting number and then counting on at a normal verbal pace, without hesitation, and gets it right each time, then I would say the student is fluently adding within 5. Reply

April 26, 2012 at 2:17 pm
I think it is a matter of how quick is quick enough. For example, in Grade 1, if a child thinks, without hesitation, “9 + 4 is 9 and 1 is 10 and 3 more is 13″ it will be quick enough to be fluent. However, it is definitely not as quick as simply recalling the fact 9+4=13.

On the other hand, Russell’s definition of fluency may be a bit problematic. For example, if a 2nd grader is adding 9 + 8 by counting on 8 times from 9, without hesitation, is he fluent? I would say no because I would want 2nd graders to be moving away from inefficient counting strategy to obtain the correct answer.

As for assessing Kindergarteners, my inclination is not to worry about “know from memory” since the CCSS does not say it explicitly. I may still use flash cards to pose questions, but I would be assessing not how quickly students give me the correct answers but how they seem to be obtaining the answers.

## Questions and answers from CCSS author, Bill McCallum

On pedagogy:Bill McCallum says:

April 13, 2012 at 10:19 am

Dear Lisa, I don’t see the standards as dictating any particular teaching method, but rather setting goals for student understanding. Different people have different ideas about what is the best method for achieving that understanding. That said, I think it’s pretty clear that classrooms implementing the standards should have some way of fostering understanding and reasoning, and classrooms where students are just sitting and listening are unlikely to achieve that.

On time and money:Primary teachers become very emotional about the placement of the time and money standards. Kindergarten- no standards on time or money; Grade 1-telling and writing time, no money standards; Grade 2-time and money; Grade 3-no money standards. I have shared my view and would like to share your response which I am certain has more credibitlity.

- Bill McCallum says:

April 13, 2012 at 10:32 amReading time and knowing the value of coins are important life skills, which students could learn in many places: in the home, in social studies, in science, in mathematics, in history, or in english language arts. There has been a tendency to overload mathematics standards in particular with these life skills, at the expense of more important work on number and operations. Perhaps this was because mathematics standards came along first, so putting these things there was a way of ensuring they were taught. The view of the Common Core is that, used in the right way, they can be tools for learning about number and operations, but they are not mathematics topics in their own right. If kids come to school with knowledge about them, or if there is a way of weaving them into the curriculum that supports the main focus, then that’s fine. But too often they become the main focus themselves. The strongest message of the Common Core is: focus on what’s important and give it the time it needs, so that kids have a chance to learn it well and progress onto other things. That required paring down previous standards.

On "Special Cases":Cobb says:

April 17, 2012 at 9:42 am

Thanks so much for taking time to repond to all of these questions. What it meant when in third grade NF 3 special cases?3NF3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

- Tad W says:

April 17, 2012 at 12:30 pmNot sure what “special cases” may be, but the denominators in Grade 3 are limited to 2, 3, 4, 6, and 8. Although the CCSS does not limit the fractions to be discussed in Grade 3 to proper fractions, there are still infinitely many fractions with those denominators. However, I would suspect that the limit of equivalent fractions in Grade 3 will be basically the proper fraction situations, that is, 1/2 = 2/4 = 3/6 = 4/8, 1/3 = 2/6, 2/3 = 4/6, 1/4 = 2/8 and 3/4 = 6/8.

Reply

- Bill McCallum says:

April 17, 2012 at 12:54 pmTad has it right. The idea is to limit to situations where you can see the equivalence by direct reasoning from the definition of a fraction, but not get into the general way of seeing equivalence. For example, you might see that 1/2 is equivalent to 3/6 using a tape diagram divided into 2 and then into 6. But you wouldn’t get into 3/6 = 3×1/3×2 = 1/2.

On Fluency:Leandra says:

April 22, 2012 at 1:51 pm

Dr. McCallum,

I need help with clarifying the fluency with addition and subtraction facts in K-2.

K- Fluent with addition & subtraction w/in 51- Fluent with addition & subtraction w/in 102- Fluent with addition & subtraction w/in 20 AND knows from memory single-digit to 9+9 (add only)

We are working on standards based report cards for our 1-2 grade levels. We have standards based in Kinder already. This year in kinder, we listed the standard as shown and then, for assessment purposes only, we used flash cards to assess fluency. (We were sure to use number talks, images, and manipulatives to ensure understanding).

When we started working on standards based for 1 and 2 we ran into some confusion, because we had at first thought in K and 1 we should be “flash card fluent” under 5 and under 10 respectively, but then in 2nd it says know from memory for the ones they should be “flash card fluent” with and the word fluent has a slightly different meaning.

What wording could be used on a report card to differentiate these skills for parents? And if the K and 1 should not be “know from memory” how should teachers assess the facts in kinder and first?

Reply

- Tad Watanabe says:

April 24, 2012 at 4:42 amIn November 2000 issue of Teaching Children Mathematics, Susan Jo Russell discussed what NCTM meant by “fluency.” She writes (p. 154):—Fluency, as used in Principles and Standards, includes three ideas: efficiency, accuracy, and flexibility.• Efficiency implies that the student does not get bogged down in many steps or lose track of the logic of the strategy. An efficient strategy is one that the student can carry out easily, keeping track of subproblems and making use of inter- mediate results to solve the problem.• Accuracy depends on several aspects of the problem-solving process, among them, careful recording, the knowledge of basic number com- binations and other important number relation- ships, and concern for double-checking results.• Flexibility requires the knowledge of more than one approach to solving a particular kind of problem. Students need to be flexible to be able to choose an appropriate strategy for the prob- lem at hand and also to use one method to solve a problem and another method to double-check the results.—I don’t know if this view is consistent with the CCSS writers’ view, but I like (yes, my personal preference) this view on fluency. It should also be noted that “fluency” seen from this perspective does not necessarily mean “quick.”

- I also think it is interesting that the CCSS distinguish “fluency within 20″ and “knows from memory ” up to 9+9. This seems to suggest that students should be fluent with calculations like 13+5 and 18-3 using their understanding of the meaning of operations and number sense. But, for 1+1 … 9+9, the CCSS seems to expect students to “just know” the facts. I also like the fact that the CCSS puts “memorization” AFTER fluency. I think if students become fluent (as explained by Russell), they will remember basic facts, too.

Reply- Leandra says:

April 26, 2012 at 8:00 amThanks Tad. So in your opinion, would you start assesessing “know from memory” addition facts only in kinder under 5 and under 10 in 1st as a progression toward the 2nd grade standard?

- Bill McCallum says:

April 26, 2012 at 1:52 pmFirst, “fluently” refers to how you do a calculation, whereas “know from memory” means being able to produce the answer when prompted without having to do a calculation. In CCSS, “fluent” means “fast and accurate.” The sort of flexibility that Tad is talking about is coded into many of the standards that are not explicitly about fluency, so it is part of the standards as a whole. I note that Tad says “fluent” does not necessarily imply “quick”, whereas I have said that it does imply “fast”. So there seems to be a bit of disagreement there, although maybe not that much; “fast” for a Kindergartner is not as fast as “fast” for a 2nd grader. If a Kindergartner adds numbers within 5 by saying the starting number and then counting on at a normal verbal pace, without hesitation, and gets it right each time, then I would say the student is fluently adding within 5.

Reply

- Tad Watanabe says:

April 26, 2012 at 2:17 pmI think it is a matter of how quick is quick enough. For example, in Grade 1, if a child thinks, without hesitation, “9 + 4 is 9 and 1 is 10 and 3 more is 13″ it will be quick enough to be fluent. However, it is definitely not as quick as simply recalling the fact 9+4=13.