Standards for Mathematical Practice: The Organization of the CCSSM Part III

This is the third post about the organization of the Common Core State Standards for Mathematics (CCSSM).

You might have noticed that when you pull up the CCSSM, the first set of standards (especially if you view the entire document as a pdf instead of on an interactive site) are the "Standards for Mathematical Practice."

The standards we've looked at up to this point have described specific content. (For example, 7.EE.A.2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.) The Standards for Mathematical Practice, however, do not describe specific mathematical content.  

Rather, these standards describe the way in which students should be practitioners of math. They describe what good math looks and feels like. 

I am fond of pointing out that while these standards describe student behaviors and understandings (because, as we talked about on Day 1, that is the very definition of what learning standards are), these descriptions of "mathematical practice" are true to anyone who really knows their way around math.

So if you lifted these eight descriptions of good math practice out of the standards document, you could also use them to describe how scientists, engineers, accountants, small business owners, CFOs, and ... well, you or I ... do math. Really, anyone who has become competent and fluent with math will be engaging in these practices.

It's worth focusing on the word "practice." The way that this term is used here is not unique to math learning standards. I think specifically of yoga. I'm a fan of yoga, and whether you're in a yoga studio or at home with a borrowed dvd from your library (I fall into the latter category, FYI), your yoga instructor is going to mention your yoga practice. This word means much more than just the repetition of yoga poses. It means everything about the way I engage with yoga ... from my mindset to my breathing to my balance to my physical ability to get into a yoga pose ... all of these things say much about my fluency and knowledge and familiarity with yoga. A yoga instructor, as a very proficient practitioner of yoga, can watch me move through a sequence of poses, and then tell you a lot about my yoga practice--whether I'm breathing right, and isolating the right muscles ... whether I'm focused and meditative as I should be, or distracted and tense.

The same is true with mathematics. The authors of the CCSSM want students to learn math in such a way that they own it for themselves--that their familiarity with it allows them to be the master--to bend the math to their will instead of being a slave to a seemingly incomprehensible and arbitrary set of steps and rules.

This kind of fluency doesn't happen overnight. (Just as I can't expect to go to two yoga classes and be able to do a kickass tree pose.) And it also doesn't look the same at every grade level. What specific actions demonstrate a fluent practitioner in 1st grade might signal a student who is struggling to keep up in 4th grade.

However, there are over-arching ideas that are consistent across grades, even if they are expressed differently at different stages of mathematical development and understanding. Precision (the focus of Practice Standard #6) in primary grades might be as simple as making a clear "T" chart ... whereas in high school, it would be more like writing out clear "let statements" so that it's easy to know what different variables are being used for in a set of equations.

The practice standards are special because of their organizational spot in the CCSSM. In former standards documents (the state standards, or the National Council of Teachers of Mathematics standards document), this type of "process standard" or "logic and reasoning standard" was given it's own domain area. Instead of putting these standards into a domain where it could be said, "Now, we're going to focus on reasoning," the authors of the CCSSM decided to make these standards overarching. The idea is that no matter what content you're teaching in the CCSSM, it should be taught in such a way that the Standards for Mathematical Practice are being encouraged, nurtured, and leveraged.

This is just as it is in my yoga class. No matter what pose I'm in, my breathing is something that needs to be attended to. ( ... as does my mental focus, my balance, and a host of other marks of good yoga practice that I have yet to learn and incorporate :) )

The eight Standards for Mathematical Practice are:

#1. Make sense of problems and persevere in solving them.
#2. Reason abstractly and quantitatively.
#3. Construct viable arguments and critique the reasoning of others.
#4. Model with mathematics.
#5. Use appropriate tools strategically.
#6. Attend to precision.
#7. Look for and make use of structure.
#8. Look for and express regularity in repeated reasoning.      

And while there is certainly overlap between them, Bill McCallum, one of the authors of the CCSSM, put together the following diagram to help us think about how they might be grouped in pairs, with #1 and #6 as overarching, being used when you use the others.

Throughout the rest of our 31 days together, I'm going to try my best to delve deeper into what each of the eight Standards for Mathematical Practice says and means. This is, by far, the most challenging part of the Common Core ... but it's also the meat of the CCSSM, and in my humble opinion, the major contribution of the CCSSM.

I'll put it to you this way: when the CCSSM first came out, and state and district math supervisors weren't sure where to start to move towards implementation of the CCSSM, the recommendation from the major professional organizations for teaching math (the National Council of Teachers of Mathematics and the National Council of Supervisors of Mathematics) told math leaders in those districts:

"Start with the math practices. If you can be teaching in a way that reflects the practices, the content will be close behind."

Are you enjoying learning about CCSSM? There are more 31 days posts here. And you can join us on facebook and twitter, for conversations between blog posts, and after October! This week's Tuesday night math chat will be on Twitter, at 9p ET--follow along using #CCSSM!

Sunday Math Task Debrief

On Friday, I posted a math task for you to check out. Hopefully you at least read it ... maybe you even pulled out a pen or pencil and wrote something out as you thought through the problem. Today, I'm sharing with you a video of me talking through the task.


      (Note: it is a small act of God that I was able to finally get this thing up for you all! From the video to the upload, to my internet connection mysteriously going away in the middle of all of it! So with that in mind, please forgive the Blair Witch inspired camera handling, and my stating a mathematical inaccuracy that "18 times 3 is 6", around minute 7 ... by the time I found the mistake, it was time to let it be good enough!)

To give a little more info, this a task that was written to reflect the standard 3.OA.A2 (third grade, in the operations and algebraic thinking domain:

Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each.

Do you see how this task relates to the standard? It's worth noticing how instead of just showing students that there are two ways to divide 18 by 3, the problem helps them discover and play with this idea.

I think that the best way to better understand math education is by doing math and thinking about how I, as a learner, engage with the math, as well as how others (my peers, my students, etc) engage with the math.

Do you like doing the math with me? I'd love for you all to share some of your kids' work, and if it fits with what I'll be talking about, I can highlight it, and even work up a little vlog post for one of our Friday Math Tasks, and Sunday Debriefs!


Are you enjoying learning about CCSSM? There are more 31 days posts here. And you can join us on facebook and twitter, for conversations between blog posts, and after October! This week's Tuesday night math chat will be on Twitter, at 9p ET--follow along using #CCSSM!

Saturday Resource

 
Since Saturdays are busy days for families with kids, I'm going to keep it short on Saturdays during our 31 days together, and just post resources.

Today's resource is from the National Parent Teacher Association. It is a series of quick, 2-page guides for each grade level. They're also available for the ELA standards ... and are available in Espanol.

To be perfectly honest, there are some key bits missing (like the Standards for Mathematical Practice--our mysterious friends I keep alluding to, but not telling you about, to keep you coming back for more--ha!), but they're a REALLY great first step to thinking about the Common Core from a parent's perspective, since the PTA is really all about connecting parents and their kids' education.

Check out these short grade-level guides here.

And don't forget about yesterday's math task! I'll be doing a follow-up post tomorrow about the problem posed, and it will be much more fun if you've had a chance to think about it a little on your own first :)

Are you enjoying learning about CCSSM? There are more 31 days posts here. And you can join us on facebook and twitter, for conversations between blog posts, and after October!

Friday Math Task

This has been an intense week with LOTS of information! I told you yesterday that I would talk about the Standards for Mathematical Practice today ... but it's Friday, and the SMPs are important, heady stuff, so I'm going to leave that until Monday, when you're fresh! I'd intended to sprinkle a little math throughout the month, and decided this might be a nice way to segue into the weekend. I'm always happy to pull tasks, but I'd also love to incorporate some of the math your kids are encountering. So if you have some math you'd like to share with the rest of us, feel free to head over to the facebook community page and post it there.

Plus, today's my birthday! To celebrate, let's do some math! Here's the task I'll pose today.

• Presley has 18 markers. Her teacher gives her three boxes and asks her to put an equal number of markers in each box.

• Anthony has 18 markers. His teacher wants him to put 3 markers in each box until he is out of markers.

1. Before you figure out what the students should do, answer these questions:
   

   What is happening in these two situations? How are they similar? How are they different?

2. Figure out how many markers Presley should put in each box. Show your work. Then figure out how many boxes Anthony should fill with markers. Show your work.

Feel free to discuss in the comments, or on our facebook community page! I plan to refer back to this task as we dig deeper into the CCSSM and some of the shifts it asks of teachers and students.

Have a great weekend!

Are you enjoying learning about CCSSM? There are more 31 days posts here. And you can join us on facebook and twitter, for conversations between blog posts, and after October!

High School Sequencing: The Organization of the CCSSM Part II

Yesterday we walked through the main organization of the CCSSM (Common Core State Standards for Mathematics), discussing where to find the document, how the standards are arranged into clusters and domains, and what additional helpful info is contained in the grade-level introductions. Today I'm going to talk a little bit about the high school standards and how they are organized, because it's not as straight-forward as the K-8 standards.

When the writers of the CCSSM sat down to write the high school standards, they had a real challenge in how to organize the high school content, because there is a lot of controversy around how to organize and sequence high school math. (I bet you had no idea there was so much drama in math ed!)

In the traditional sequence, the courses are Algebra I, Geometry, and Algebra II.

In the integrated sequence, the same content is covered, but in a more ... well, as the name says ... integrated manner. Integrated courses are referred to more generally as Math I, Math II, and Math III.

The controversy and preference for one sequence over another has been around for awhile. (I am a strong supporter of the integrated sequencing, personally.) Some states have traditional sequences, some integrated, and some states leave the choice of sequence up to districts.

Since a goal of the Common Core was to be ... well ... in common among many states, the writers needed to come up with a solution that would keep everybody happy. (Keep in mind, the two sequences cover the same materially, they just group content differently, and their chronology for covering it is different.)

As the peace-makers that they are, CCSSM authors opted to code the high school content by domains, and then offer suggested pathways for covering that content. (There are actually four suggested pathways for covering the content, because the authors also included ways of bringing the high school material into middle school, as a number of states and schools have been pushing to do with the algebra-for-all movement and others.)

So let's cut to the chase, shall we?

In Appendix A of the CCSSM, you can find these pathways. (When you're on the interactive standards page of corestandars.org, Appendix A is in the menu at the left ... you just have to scroll really far down. It's at the bottom.)

I've pulled up the traditional pathway to give us a concrete example to discuss:

The courses are listed across the top, with the domains listed in the left-most column. Let's look at the domain of "Seeing Structure in Expression" from the Algebra topic in HS. If we look at the Algebra I course, we see that there are six standards from this domain present:

• A.SSE.1a
• A.SSE.1b
• A.SSE.2
• A.SSE.3a
• A.SSE.3b
• A.SSE.3c

You'll notice that the coding is a little different. There isn't a grade-level designation embedded in the code. Instead, there's the HS topic. In this case, that topic is "Algebra" (which should not be confused with "Algebra I" or "Algebra II" which are courses). However, once you shake you head a little to clear the fog around a topic called "Algebra," you notice that the next part of the code is much like it was for the K-8 standards: SSE refers to the domain: Seeing Structure in Expressions. The last bit of code refers to the standard number and cluster under the SSE domain.

You might also notice that some of these standards are also present in Algebra II, but where they apply in Alg I to linear, exponential, and quadratic expressions ... now in Alg II the same expectations are applied to polynomial and rational expressions.

If you have a kiddo in Algebra I and you wanted to see what standards outline their goals for the year, you would read down the Algebra I column, through the appendix, to see what content is there.

Caveat: since the CCSSM did not mandate a sequencing, your state may have kept the same content overall, but done a little shuffling. Therefore, I strongly suggest that you check with your child's teacher as to what they reference for their standards--the CCSSM document, or a more specific state document.

Tomorrow we're going to talk about one more structural thing--the Standards for Mathematical Practice. In my humble opinion, the Standards for Mathematical Practice are the STARS of the CCSSM, and as many stars often do, they've caused a bit of drama. Come back tomorrow to learn more :)


Are you enjoying learning about CCSSM? There are more 31 days posts here. And you can join us on facebook and twitter, for conversations between blog posts, and after October!

Nuts and Bolts: The Organization of CCSSM Part I

Hey! You came back! Great! Hopefully yesterday's info-heavy post didn't overwhelm you, but just helped us to set some groundwork and common definitions to work from :)

Today's going to be a quick primer on navigating the CCSSM (Common Core State Standards for Mathematics) document, so that as a parent, you too can read and know what the standards say and what goals are laid out for your kids' education.

First things first: did you know that the CCSSM is available online, for you (yes, you!) to read for yourself? Just pop on over to corestandards.org and you can read both math and ELA standards documents, as well as read up on other info about the standards, what states have adopted, and who has made a public show of support.

Once you're there, you'll see a button for the "Mathematics Standards" (the arrow below is pointing to it). Go ahead and click it.

You'll be brought to this nice interactive page where you can navigate through the standards by intro, standards for mathematical practice*, and grade level. First check out the info under intro--that's what we're chatting about today.

You'll notice that once you click "Introduction," some broad information comes up.

The standards' authors put this bit in as a guide map so that you could know how the standards are organized and how to navigate them. The three organizational levels (in increasing magnitude) are individual standards, clusters, and domains. I've copied and pasted the descriptions below. (But I would be remiss if I didn't tell you that it's also very much well-worth the read to go ahead and read those three paragraphs below the structural info as well. Those three little paragraphs explain how the writers of the standards see the standards in the larger scheme of education--not as a curriculum, and not as a prescriptive way to teach, or chronology to teach--but as I described yesterday--as a backbone to give math education its form and function.)

Standards define what students should understand and be able to do.

Clusters summarize groups of related standards. Note that standards from different clusters may sometimes be closely related, because mathematics is a connected subject.

Domains are larger groups of related standards. Standards from different domains may sometimes be closely related.

A few things that are worth noting here: there's a nesting happening. I've drawn up a little diagram to help you see this. A grouping of standards is called a cluster. Some clusters are tiny and only have one standard in them. A grouping of clusters is called a domain. Both domains and clusters are formed by standards that have related content.

Domains stretch across grade levels. Clusters do not. Not every domain is present in every grade. Counting and Cardinality, for example is really about establishing the ability to count, and to understand quantity. It makes sense that this domain would begin and end in kindergarten, and wouldn't have a need to spread all the way to 12th grade.

The graphic below maps out which domains exist at which grade levels, with the grade levels across the top and the domains listed below. If you read it left to right, it gives you a sense for which domains lay the groundwork for later domains (for example, Measurement and Data lay the groundwork in the elementary grades for Statistics and Probability in the middle grades).

Let's take a look at a specific example from the CCSSM to help us see how this organization works. I've got pulled up the Grade 1 Standards, and have chosen the "Measurement and Data" domain from the menu that appears below "Grade 1."

In 1st grade, in the Measurement and Data domain, there are four standards. These standards are organized into three clusters. Each standard has a code that tells you exactly where it can be found in the standards. If we take, for example,

CCSS.Math.Content.1.MD.B.3

If I read the code left to right, I can tell you that that standard is a common core (CCSS), math (Math), content (Content) standard, that it occurs in first grade (1), is a measurement and data standards (MD), is part of the second cluster in the MD domain for 1st grade (B), and is the third MD standard overall for 1st grade (3).

Your state's coding may be slightly different, because as each state adopted the Common Core, they needed to fit it into their state's existing structure as much as they were able. Though in my experience, most states' coding is very similar to this, or easy enough to decode once you understand the structure of standards, clusters, and domains.

It's worth noting as you look through this particular excerpt of the CCSSM that not every standard is the same "grain size" or necessarily takes the same amount of time to teach. Some standards are more complex and have multiple pieces. Some are fairly straightforward, like 1.MD.B.3.

And a last element of the standards that you will really want to look at: the grade-level introductions. These introductions give you a sense for what your child will learn through the course of the year. They also give teachers ideas for what areas are most key to spend instructional time on, so that they know which concepts are foundational for future grades, etc.

If, as you peruse your kids' grade levels, you feel like this language is heady--you're absolutely right! And that's ok! Standards documents are not for children. And they're also not for novices. This is the blue-print of K-12 math education--it is complex and sophisticated and deserves a level of language precision that reflects that. Keep in mind--this document wasn't written with parents (and the general populace) as the intended audience. Rather, it was written by mathematicians and math educators to communicate a plan for K-12 math education to other math educators--educators who write curriculum, and administrators who map out curriculum, researchers who study math education, and teachers who enact these sophisticated and complex processes.

I promise that I didn't bring you to the CCSSM document to make you feel dumb :) I brought you here to demystify the standards--to show you that you have the ability to search and read what's actually in these documents for yourself. Hopefully you feel like you have the tools to navigate these documents on your own.

* You might have noticed that I just so happened to not mention the Standards for Mathematical Practice that are at the top of the menu (above the Introduction), and are listed at the end of each grade level introduction. That's because I'm going to spend a whole post talking about what those standards are (spoiler alert: they're SUPER DUPER special!), and plan to dig deep with you through this series, so that we can better understand them together.



Are you enjoying learning about CCSSM? There are more 31 days posts here. And you can join us on facebook and twitter, for conversations between blog posts, and after October!

What is the Common Core State Standards for Math?

It seems appropriate that for the first day of our 31 day series, we begin by first discussing what the Common Core State Standards for Math is.

First, let me explain what state standards are, because they're not new. An excerpt of wikipedia's definition:

Learning standards (also called academic standards, content standards and curricula) are elements of declarative, procedural, schematic, and strategic knowledge that, as a body, define the specific content of an educational program. Standards are usually composed of statements that express what a student knows, can do, or is capable of performing at a certain point in their learning progression (often designated by "grade" or its equivalent).

(Emphasis added by me.) I would add that these standards are adopted by state legislatures as a means of structure and accountability to ensure that all students in the state are learning the same content at the same grade level. You can think of standards as a list of goals for teachers and students.

As a list of goals for all teachers by grade level and subject, standards then serve as the backbone for other elements of education. How should textbook companies and other curriculum developers decide what to put in their materials? Well, if there's a set of goals for third grade social studies in New York, then the curriculum company can make sure that their materials are helping teachers and students in New York achieve those goals, and can then say that they are "aligned with the NY standards." (FYI, in education, being aligned with the standards is a BIG deal. If your program or book or whatever isn't state-standard aligned, it's a non-starter. Educators know that )

I personally, like to think of standards as the skeleton. Without them, education would be like an ameoba, taking on different shapes, being moldable and malleable. Instead, standards give a definite form to shape, size, definition of what education looks like at the K-12 level.

But the skeleton illustration doesn't end there, because just as a human is not solely their bones, education is not solely the standards. Consider the following graphic:

Standards are the base from which we develop and shape curriculum (text books, handouts,  discussion questions, etc). But you can develop many different curricula that all meet the same standards. All you have to do to understand this is to go to an education conference, where the exhibitor hall is guaranteed to be occupied by a number of textbook publishers, who each will assure you to be "Common Core Aligned," though they do not handle the topics the same way.

Standards also inform instruction. If a particular topic is in the standards, a teacher is held accountable for teaching that content to their students. However, the standards do not dictate how that content is learned. One teacher may opt to begin with an activity that explores related concepts and helps students understand the need for this new concept (for example, if the new concept is the notion of negative numbers, a teacher may have students explore monetary transactions, and think about what happens when you want to buy something from a friend, but don't have enough money--how do we understand the debt? How can we represent it on paper? etc.) Another teacher might introduce negative numbers by drawing a number line on the board and asking students if there's anything to the left of zero. A third teacher might draw on students' notions of negative temperatures, and ask them to explain what it means for it to be " - 4 outside."

And lastly, standards are the basis of assessment (tests). I'll talk a good deal more about assessment in other posts, but standards and assessment go hand-in-hand. Or at least they should. Standards stake out the goals, and assessments check whether those goals have been reached. Assessments, like standards, are determined at the state level.

Just as we all have the same skeletal structure, all classrooms at the same grade level in a single state have the same learning standards. However, local choices made at the district and school level determine what the curriculum looks like. Even more local choices made at the teacher level determine what specific lessons look like.

It is a common fallacy that common standards mean the same lock-step education for all kids.

The reality is that standards have been around for several generations now. The reason these standards are getting a great deal of attention is because it is unprecedented for different states to have a common set of standards. Because education is a state's right (not a federal one), each state has historically gone about determining what should be learned at each grade level on their own. But in this case, many states decided that it was time to begin working together, and thus to adopt a common set of standards, that translated across state lines, and allowed for an economy of scale. (More on how this sudden cooperation and collaboration came about in a later post!)

So to recap, the Common Core State Standards

• is a document that outlines learning standards in math and English language arts
• have been adopted by individual states via their usual adoption method (for many states, via the legislature)
• are not a curriculum
• do not dictate how topics should be taught
• do not dictate how an individual teacher teaches
• are ground-breaking because for the first time, states share common learning goals
• are ground-breaking because they allow for collaboration across state lines

This is a lot of information, (thanks for hanging with me today!) but I did want to lay a foundation for us, that cleared up a number of the common misconceptions or misinformation that keep floating around the internet :)

Tomorrow, I'll talk about the organizational structure of the Common Core State Standards for Math (CCSSM), so that you, as a parent, can dive into the document yourself, and take a look at the kind of stuff that it says about what your kid will be learning at their grade level.

PSST!! Have questions? Want to talk about this more? You're in luck! Tonight, on the facebook page, I'm hosting a Common Core Math for Parents math chat at 8p ET! Hop on over, like our page, and join the conversation! Bring a question, or even a picture of a homework problem that's got you stumped--we're here to chat!