Here is the draft progression for the Kindergarten Counting and Cardinality Progression and the K–5 Operations and Algebraic Thinking Progression. It includes some edits to the K–2 progression from the previous version: ccss_progression_cc_oa_k5_2011_05_30. [File updated 6/1/2011.]

[29 July 2012] This thread is now closed. You can ask questions here.

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## About Bill McCallum

I was born in Australia and came to the United States to pursue a Ph. D. in mathematics at Harvard University, met my wife, and never went back. I am a professor at the University of Arizona, working in number theory and mathematics education.

The attachment is very comprehensive, and I find the layout rather easy to follow. Thank you for taking extra time to post it and share. It’s helpful.

Kind regards,

Tracy Watanabe

Based on the experiences today and the reading, I can see that a lot of emphasis on these new standards are going towards how the students will be getting solutions more than if they are getting solutions. In the reading, it shows the different levels of thinking that it would take for a student in 6th grade to solve a problem as well as a 7th or an 8th grade student would. As the child gets to the higher-grade levels, the complexity of solving a problem deepens. The child must start to think more abstractly as the level increases. The student must also be able to model their learning in more complex levels as their level increases. The main thing that I am going to take from today’s experiences and readings is that we as teachers cannot just show the students how to take a method and use it every time as the only way to solve a problem. We have to be willing to let the student “model” and demonstrate how they were able to solve the problem so that students can see as many possible ways to approach a problem as opposed to just doing it the teacher’s way. I think this is a great way to get the students involved in their own learning.

Can you clarify 5OA1 specificially states use parentheses, brackets, or braces in numerical expressions….but in the progressions text pg 32 draft 5/29 it is stated that “expressions should not contain nested grouping symbols.” We are confused. These two statements seems to contradict each other. Thank you in advance for your reply.

“Nested grouping symbols” means grouping symbols within grouping symbols, like 5[3 + 2(9-2)]. The progression suggests limiting 5.OA.1 to expressions without nested grouping symbols, such as 3(4+5).

Thanks for your response. To further clarify the use of parenthesis, brackets, or braces would this expression be inappropriate for 5th grade 2{5[12+5(500 – 100 )+ 399]} ? We are thinking this would be inappropriate due to the nested symbols. Is there a situation where parenthesis, brackets, or braces would be used where the symbols are not nested? Can brackets or braces be used without parenthesis? Thanks for your help.

I was fascinated with grade 6 – ‘Apply and extend previous understandings of arithmetic to algebraic expressions’. As a high school algebra teacher, I was interested in how the middle school students are introduced to use of variables in word problems. When some of my students show reluctance in starting a problem, I approach it in the same manner – start with one or more numerical expressions for the same given situation, then replace the number by a variable.

The progression of ccss is smooth and the grade 8 prepares them well for the higher level algebra in high school.

Thank you for a very comprehensive document. What is your timeline for publishing for ccss for high school?

Sincerely,

Meera Manghnani

Thanks for the comments. In answer to your question about the high school progressions, they will be coming out during the fall.

Do you have any information about the results of the Mathematics Curriculum Analysis Project, funded by the Brookhill Foundation and Texas Instruments, and supported by CCSSO, that was to provide a set of math curriculum analysis tools to DOEs by June 1, 2011?

We’re working on aligning our current Kindergarten units with the common core. The progression documents have been very helpful. We are struggling with the difference between K.CC.4 and K.CC.5.

The two are complementary: K.CC.4 describes an understanding and K.CC.5 describes a performance. So, for example, you might show a student 5 teddy bears and ask the student to count how many there are. The performance of counting and saying how many is K.CC.5. But, after a student has given a correct answer, you might say: “show me the 5 teddy bears.” Some students will point to the last teddy bear counted, or count them out again, showing they have not yet connected counting to cardinality (i.e, the last count tells how many). A student who has acquired this understanding might gesture to the entire collection of teddy bears. This distinction is explained more in the NRC 2009 report on early childhood learning, which influenced the writing of the Kindergarten standards.

On the same topic, I have a group of K teachers wanting to assess the standard KCC5 by having the children count a printed picture of 7 objects in a circle and 18 objects in an array. I think the intent is to have children, when faced with real objects in these configurations, count with a plan to improve accuracy and avoid recounting. The assessment then would be having them count real things and move objects as needed for them to be accurate. Am I interpreting this correctly?

Page 6 and 7 are the “Overview of Grades K-2” for the OA domain. Is there an overview for grades 3-5 somewhere?

On page 18 you say, “The word fluent is used in the Standards to mean ‘fast and accurate.'” This definition typically uses the words “accurate, efficient, and flexible.” Was “flexible” left off for a reason?

The next sentences go on to say, “Fluency in each grade involves a mixture of just knowing some answers, knowing some answers from patterns…, and knowing some answers from the use of strategies.” This certainly seems to indicate that flexibility is important to the authors, but it is noticeably absent from the definition. Please explain?!

Thanks,

Brian

The definition “fast and accurate” refers specifically to how the word “fluently” is used in the content standards. For example, 2.OA.2, “Fluently add and subtract within 20 using mental strategies …” means “rapidly and accurately add and subtract within 20 using mental strategies …”. This does not exclude all the things referred to in your second paragraph as a description of how fluency might be acquired, what mental processes might lie behind it, or to what uses it might be put.

And the specific use of the term in the content standards for the purposes of being clear about what is meant does not exclude a broader understanding of the term generally. In fact, the standards cite the Adding it Up definition of procedural fluency (“skill in carrying out procedures flexibly, accurately, efficiently and appropriately”) on page 6, in the preamble to the standards for mathematical practice.

Thank you for your quick reply.

I always worry about what a document like this might unintentionally communicate by leaving off something like “flexibility;” however, as I went back and looked at all of the “fluency standards” I think I saw your point…

For some of these standards it would be appropriate (and highly desirable) to ask for flexibility (K.OA.5, 1.OA.6, 2.OA.2, 2.NBT.5, 3.OA.7, 3.NBT.2, and 7.EE.4a).

However, other standards specifically require one particular algorithm, which means asking for flexibility would be inappropriate (4.NBT.4, 5.NBT.5, 6.NS.2, and 6.NS.3).

Because of this, the easiest way to clearly define “fluently” for all of the content standards is to leave off “flexibly;” but this is not intended to undercut the importance of “flexibly using appropriate strategies/algorithms,” when it fits within the standard (which is communicated, as you noted, in Adding It Up and MP5). Is this correct?

Basically yes, but the phrase “use X flexibly” is liable to cause confusion because it is ambiguous. It makes sense to want students to be flexible in their decisions about when to use an algorithm; for example, we don’t want them to blindly multiply 40 x 70 using the standard algorithm. But algorithms in themselves are, by definition, fairly rigid. So, given that a choice to use a particular algorithm has already been made, there’s no value in having the same student do it a different way every time; the value of an algorithm is in part its regular, repetitive nature. On the other hand, we want teachers to be flexible about the choices they allow different students to make, e.g., how they notate a particular algorithm, or whether they choose to use one at all for a given problem. The phrase “use algorithms flexibly” could apply in all these situations and others I haven’t described; sometimes you want it and sometimes you don’t, depending on the meaning.

This now makes perfect sense to me. Thank you.

I don’t know if this is the right place to post this question, but I couldn’t figure out anywhere else to go. Where does patterning and skip counting fit in to the common core? Many kindergarten teachers feel that teaching the patterns using various attributes is a prerequisite for skip counting, but it isn’t even mentioned in their standards. Many 1st grade teachers are asking why it was taken out of their standards because they feel it is a prerequisite as well. My 2nd grade teachers have asked me if they should skip count by any number starting at any number or only teach skip counting by 2, 5, adn 10 starting at 0. I am unclear on how to answer many of these questions. I have tried to read the Learning Progressions about patterning, but that left me with more questions since the Patterns, Relations, and Functions (on page 28) left me with a bigger question mark. I would just love to hear some clarification on this topic. I really look forward to hearing back from you and appreciate everything you are doing!

Jackie, could you tell me what you mean by the Learning Progression on patterning? There isn’t one on this website. Reading the progressions on Operations and Algebraic Thinking and Number and Operations in Base Ten given here might help your question.

Patterning and skip counting can support the work of learning to count and add whole numbers, but they can also be used in ways that don’t support that. For example, given a repeating patter red, blue, blue, red, blue, blue, … a teacher could ask what the next color is, or could ask questions that get more at the underlying operations of addition and multiplication (not in Kindergarten, obviously). For example, you could ask about the size of groupings, how many groups it takes to get to 12, what the 23rd color would be, and so on. The same goes for skip counting: if it is connected to addition and multiplication, it can be useful, but if it just a matter of memorizing a sequence, then it could get in the way of understanding counting as cardinality, and understanding going to the next number as adding 1. Skip counting by 5 or 10 can reinforce base 10 understanding, because you notice the pattern in how the digits go up (and this includes starting from a number other than 0). Skip counting is not a goal in its own right, however. In short, both skip counting and patterning are viewed as supporting learning of operations and their properties, rather than as being learning objectives in their own right.

Bill,

Thank you so much for all of your hard work and for responding back to me. This helped a lot. The progressions I was referring to are titled “Learning Progressions Frameworks for Mathematics K-12”. Here is the link to that document:

http://www.nciea.org/publications/Math_LPF_KH11.pdf (I actually didn’t realize this “LPF” document wasn’t part of your writings.)

Thank you, again. Jackie

I am hoping you can clarify for me a statement that is made on page 29 in the second paragraph, “…note however that multiplying by a fraction is not an expectation of the Standards in Grade 4.” It seems like this statement is in direct conflict with Standard 4 in the “Number and Operations – Fractions” domain. Does this statement really mean that it is not an expectation in Grade 4 for students to be able to identify multiplication compare problems as problems that involve multiplying by a fraction? Please help me identify what information I am missing here. Thank you!

4.NF.4 says “Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.” Multiplying a fraction by a whole number can be understood as an extension of multiplying a whole number by a whole number. Just as 5 x 3 is seen as 5 groups of 3, 5 x 1/3 can be seen as 5 “groups” of 1/3 (portions might be a better word in this context).

On the other hand, the OA progression is referring to multiplying a number by a fraction, which is conceptually more difficult. Thus in Grade 5 we have 5.NF.4, “Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.”

Thank you for the quick reply. I appreciate the help in identifying where my misunderstanding was. Also, thank you for writing and sharing the progression documents. They are extremely helpful.

Pg. 21 of this progression document — in the left-hand column says

“(by the end of grade 2, students know all sums of two-digit numbers from memory)”

this does not correspond correctly to 2.OA.2

Oh, good catch, thanks!

Hello,

As a group of us was reading through this progression, it was noted that on the bottom of page 29, in the last sentence, it says, “but the number of steps should be no more than three and involve only easy and medium difficulty addition and subtraction problems.” Since 4th grade students are supposed to solve multi-step problems with all four operations, is there some similar guidance to follow if you are including multi-step problems with multiplication and division?

Thanks for any help!

Dianne

I think this is just an oversight in the Progression, and should apply to multiplication and division as well. Thanks for pointing that out.

Hi Bill:

First of all, thank you, for all this work. I am a big fan of CCSSM and one of my personal favorite is this Domain, CC/OA. In looking at its progression I was hoping you could clarify a couple of things for me.

K.CC.6 asks that students make the assessment of qualifying the group as greater than, less than or equal to, yet, in many cases, it may be assessed the other way around, where the student selects the group that is greater or lesser. Or one could ask a student to look a set of 4 or 5 groups and select the 2 that are equal. Are these things still within the boundaries or expectations of the standard?

I guess my general question is how far are we to interpret a standard? Should it be taken strictly at face value or can we make assumption that if written one way, the opposite or the inverse should also be true? Some examples: if students are to represent and equation with objects should they be able to write an equation from a representation; if students are asked to decompose should they be able to compose; if they are asked to sequence numbers in a certain order they should be able to describe in what order numbers are sequenced, etc. And what about combining standards into question that may be more complex? For example by combining 1.OA.7 & 1.OA.8 you could ask students to find the unknown to make 4 + 8 = ? + 6 true.

My other question is regarding the definition of fluency as “fast and accurate”. How does process vs. speed play a part of it and how would you distinguish ultra efficient mental strategies from just memorization?

Lastly, do you know how soon may we expect the geometric measurement progression and the geometry progression to come out?

Thanks again,

Ivan

First, I think your ideas for assessing K.CC.6 are quite reasonable, and well within the scope of the standard. For your general question, wouldn’t make it a general principle that you can always assume the reverse of an operation is implied, but the two examples you mention seem reasonable to me. And it’s certain reasonable, in fact is a good idea, to think of assessment and instructional tasks which assess more than one standard at a time, and your example about combining 1.OA.7 and 1.OA.8 is nice.

On your fluency question, I’m not sure I understand the first part (process vs. speed), but I think “ultra efficient mental strategies” certainly have a role in fluency. To this day see 8 + 7 = 15 as 8 + 7 = 8 + (2 + 5) = (8 +2) + 5 = 10 + 5 = 15. I sort of see the 2 flash over.

The measurement progression is almost ready, the geometry progression is taking a bit longer.