Where's the math?

In this final post on the Five Practices, we look at the final Practice - Connecting. Previous posts from this series explore preservice elementary teachers work on Anticipating, Monitoring, Selecting, and Sequencing related to this lesson from Context for Learning Mathematics (CFLM - Fosnot et. al.). Because elementary children are in short supply at our university, the future educators use professional development materials gathered from a third-grade classroom working on the turkey-cost problem as a substitute.

Image used with permission of authors 
New Perspectives on Learning

In this part of the exploration, my teachers watch video of the third-grade teacher wrapping up the lesson. They look for evidence of the teacher Connecting (1) the various approaches used by the students, (2) key mathematical ideas, and (3) a new, related problem. Connections between the students' approaches are especially strong given choices she made in Selecting and Sequencing the work to share. 

The transition to the next problem also seems well-thought-out as students determine how long to cook the 24-pound turkey if the cookbook suggest 15 minutes per pound.

Image used with permission of authors 
New Perspectives on Learning

The connection to the turkey-cost problem is obvious to the teachers.

The part of the Connecting sequence that seems most lacking is Connecting the students' work to key mathematical ideas. I try to make it clear that it is unfair to criticize the teacher because we do not know what happened before or after the lesson or, for that matter, other contextual factors that might have influenced her instructional decision making. However, to ensure that the teachers recognize the big ideas associated with the different approaches, I offer the following additions to the lesson.

The first three students are applying the Distributive property to the cost per pound decomposed into it's whole number and decimal parts. In order to compute 24x0.25, the students are essentially factoring 24 into 6x4 (4 being the multiplicative inverse of a quarter, 0.25) and then using the Associative property to regroup the order of multiplication. Consequently, the original expression, 24x1.25, simplifies to 24+6 or $30 - the answer to the turkey-cost problem.

The last pair of students used a more efficient approach that bypassed the need for the Distributive property. They still factored 24 into 6x4, but then they used the Associative property to multiply 4x1.25 first, to get 5, and then 6x5. Again, the answer is $30.

While some of the teachers think the key mathematical ideas presented in these series of expressions might be beyond the third-graders' current understanding, the Associative and Distributive properties are found in the Grade 3 Common Core State Standards (3.OA.B.5).

What do you think? Is this too much to expect of third-graders? Do you have another way that these key ideas might be connected to the students' work?

How did the teacher organize the turkey-cost discussion?

My preservice teachers have reached stage four, Sequencing, of the Five Practices (see previous posts for Anticipating, Monitoring, and Selecting). Because these teachers have no direct experience facilitating a discussion involving students reflecting on their mathematical thinking, we return to the third-grade class and watch a video of how the teacher Sequences the work of the students on the turkey-cost problem. I ask the future educators to watch the consolidating conversation and hypothesize why the teacher decided to order the student-work the way she did.

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All images are from New Perspectives on Learning
used with the permission of the authors

Emma and Emma take the $1.25/pound price and start with a friendlier number - $1 per pound. If that were the price, then the 24 pound turkey would cost $24. But they recognize they need another twenty-four $0.25 to find the total costs. So they count by 25s, keeping track of how many 25s they have counted underneath. They find they need another $6 for a total cost of $30.

Harry and Ese use a similar strategy of breaking the price per pound into a dollar and a quarter. However, instead of counting by 25s, they gather the quarters in groups of four. Each group of four quarters is a dollar. There are six groups. Therefore, $6 must be added to $24 to get the total cost of $30.

The next pair, Nellie and Nate, also start by taking off the 25 cents to get an initial cost of $24. Then they group the quarters, but they do it differently than Harry and Ese. Instead of showing "pictures" of quarters, they use a table to represent the relationship between pounds and dollars at $0.25 per pound. The table shows that 24 pounds requires an extra $6. Again, the total cost is $30.

Finally, Suzanne and Rose share a unique strategy that does not break up the $1.25. They know that "4 pounds is 5.00" and use this to jump by 5s on the open number line. There are six jumps because there are six 4s in 24 pounds. Although they use a different approach, Suzanne and Rose also find the cost to be "30 $ in all!"

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After observing the Sequence of approaches, the preservice teachers offer their hypotheses about why the teacher put the student-work in this particular order. Some suggest it might have progressed from "most popular" to "most unique." Others think it was based on increasingly sophisticated structures. A few wonder if it might be related to the different representations being used.

Why do you think the teacher ordered the work in this way? And where do you think the teacher goes next to complete the last Five Practices stage, Connecting? As always, your participation in the comments is appreciated.

Who should share their turkey-cost solutions?

All images are from New Perspectives on Learning
used with the permission of the authors

In preparing future educators to facilitate productive math lessons, we provide opportunities for them to apply the Five Practices to problems like the one shown here. Teachers began (in the first post of this series) by identifying a Standards for Mathematical Practice (SMP) to focus on and Anticipating student solution strategies related to this SMP. Next (in the second post), they watched video of third-graders solving the problem in order to Monitor their efforts and compare them to the Anticipated responses. The teachers were also given copies of the students' work to examine. Teachers use this work to begin the process of Selecting who might share during a whole class discussion.

For example, the teachers focusing on SMP 5 might Select these students to share because they all used the open number line as a tool.

The discussion could revolve around how this tool was used appropriately and strategically.

Another group of teachers, focusing on precision could Select the work of the third-graders shown below since they seemed to arrive at different answers.

The class could work together to determine what question each pair of students was answering and how to move toward a correct solution.

Finally, teachers wanting to focus on structure might Select these four student-pairs.

This work included different ways students used the structure of money (decimals) to arrive at a solution.

By strategically Selecting student work to share, a teacher does not leave the  ensuing discussion to the vagaries of volunteers. Consequently, the resulting classroom conversations becomes more purposeful and productive. But first, the teacher must apply another of the Five Practices - number four, Sequencing. 

How might a teacher organize the Selected student-work for each SMP in order to maximum learning and why?

How did those third-graders determine the turkey's cost?

All images are from New Perspectives on Learning
used with the permission of the authors

Having anticipated third-graders' thinking for the scenario provided above (the first of the Five Practices which was attended to in the first post in this series), my preservice elementary teachers are ready to engage in the next Practice - Monitoring students' thinking. Unfortunately, the university has yet to meet my request for a lab school which means that elementary-aged children are in short supply in my classroom. Given my desire to create as authentic experience as possible for my teachers, this creates a problem.

Luckily, Dr. Catherine Fosnot and her colleagues have gathered classroom videos and student-work from elementary kids working on problems from their Context for Learning Mathematics (Fosnot et. al.) series (including from the turkey cost lesson). While it's not the same as monitoring actual students, it does represent the same experiences inservice teachers might have in Professional Development (PD) sessions using Dr. Fosnot's materials.

This PD involves helping teachers to develop phronesis. "Phronesis is situation-specific knowledge related to the context in which it is used—in this case, the process of teaching and learning." (from p. 147 of Young Mathematicians at Work: Constructing Multiplication and Division) By watching the video and examining the students' work, teachers are able to observe an authentic lesson and reflect on the teaching moves that support students who are immersed in doing mathematics.

The preservice teachers in my class watch the video and observe the students finding the turkey cost. Just as many of them predicted, the third-graders are splitting the $1.25 per pound into a dollar and a quarter. Students' papers (like the one on the right) show that they understand the 24-pound turkey will cost $24 plus 24 quarters. Pairs of students use a variety of approaches to determine the total cost of the turkey. As my teachers review the third-graders' efforts, the teachers move toward the next phase of the Five Practices - Selecting students to share their work based on the Standards for Mathematical Practice (SMP) selected at the beginning of this process. 

We will continue this work in the next post. But first, which SMPs would you say the work of Emma and Emma highlights?

How much is that turkey in the window?

Disclaimer: I receive no financial benefit for my endorsement of the Context for Learning Mathematics (CFLM - Fosnot et. al.) curricula or the associated professional development resources shared in this series.

In my efforts to make my classes for preservice elementary teachers more accurately reflect the work of teachers, I try to craft my lessons as professional development sessions. We spend a significant amount of our time applying the Five Practices for Orchestrating Discussions to activities from established K-6 mathematics curricula. Recently, we used a lesson from CFLM's The Big Dinner unit in anticipation of Thanksgiving.

From Professional Development Resources
used with permission of author

This introductory lesson asks students to find the cost of buying a 24 pound turkey. My teachers start by selecting a Standard for Mathematical Practice (SMP) as a goal for the lesson. (We could also look at content goals, but these teachers need more practice with the SMPs.) Having chosen a goal, the teachers begin applying the First Practice: anticipating possible student responses associated with the goal. The students in this scenario are third graders who are unlikely to use the standard algorithm for multiplying decimals.

Many of the teachers anticipate that the students will break the $1.25 into dollars and cents. For the teachers who identify "Look for and make use of structure" as their SMP goal, this prediction seems reasonable. Some of the teachers consider the models and tools (SMP 4 and 5) students will use to solve the problem. Other teachers, who are attending to precision (SMP 6), wonder where students might make mistakes in their computation. Nearly all the teachers are interested in the different strategies the students will use to solve the problem.

Before we move on to the Second Practice, monitoring students' work, I want to give you an opportunity to add the SMP you would choose to focus on for this problem and possible student responses you might anticipate. As always, please add your contributions to the comments.

In the next post, I will explain how preservice teachers can carry out the remain Practices of Orchestrating Discussions even though they are not actually in an elementary school classroom. 

What is this craziness?

From 1075 KZL Facebook Page

If we are to believe memes like this, the math currently being taught in schools has never been so meaningless. What exactly is this craziness? According to a recent NBC News story, it involves using number lines.

I understand that this approach can be confusing to adults who were taught an algorithm without really understanding why it works. Unfortunately, many of these adults have forgotten how memorizing rules without reasons made them feel as a kid.

As students, we wanted our mathematical efforts to make sense. But instead, we got little ditties like, "Mine is not to question why, just invert and multiply." So we made up that doing math simply meant following certain rules given by some authority.

It is reasonable that adults who grew up with this view of mathematics might be confused by instructional approaches meant to foster the development of mathematical understanding rather than rote rules. However, my primary concern is the confusion of the kids not the adults. To do otherwise is not only crazy, it is insane.

from 1Funny.com

Will it fit?

I used this lesson in MTH 221 (Mathematics for Elementary Teachers) to address Common Core State Standard 7.G.B.4. It seems to have a lot of potential, but there are still some elements that I think need to be tightened up. These are written in red - along with some other thoughts. I would appreciate any feedback on how to improve this lesson.

[Schema Activation]

How many of you know your fitted hat size? For example, I wear a seven-and-seven-eighths. Today, you are going to find your hat size and what that number means. 

I like the DC cap for a couple of reasons. First, obviously, DC are my initials. Second, it foreshadows the circumference and diameter relationship we will explore in the lesson.

In the past, I have had students measure a bunch of circles to find the ratio between circumference and diameter. It has been a struggle to make it an interesting lesson. This connection to something personal (hat size) seemed like it might be an improvement.

According to LIDS.com, there are a couple of ways to determine your hat size.

We will use both a flexible tape measure and their printable ruler in order to...

[Focus]

... consider the following questions:

• How are hat sizes and head-measures related? (In other words, if we didn't have access to their table, can we determine a hat size given a head-measure?)
• We are supposed to be working on CCSSM 7.G.B.4. Is this connected in some way to circles? Too obvious?
• Why don't hat makers just use the head-measurement as the size?

[Activity]

Measure your head using both the flexible tape measure and the printable ruler.

Feel free to wear the printable ruler as a stylish headband as you work. Optional

Place both your head-measure and hat size on a sticky note and place it at the proper coordinates on our graph.

What does our graph show? Is there a relationship? If so, what do you predict the relationship to be? (If we input 22 inches for head-measurement, what hat size is the output? What if we input C inches?)

Here are a couple of tables for hat sizes from Lids.com. Let's use them to see if we can determine the input-output rule they are using to find hat size from head-measurement. 

There are other hat size tables (for example), but I like that this one makes 22 inches a hat size of 7 because 22/7 is often used as an approximation of pi.

[Reflection]

What did you find? What is the rule?

Des found the following: y = 0.333x - 0.333. Does it work for our table? If it's correct then what hat size does a person need for a head-measurement of 24?

Okay, I wanted to play with the new Desmos linear regression feature - sue me. Is it a problem that this line doesn't go through the origin? That the slope actually represent an approximation of 1/pi? During the lesson, it seemed like this portion required a lot of scaffolding.

So what does the hat size mean? Let's take our headband and place it on the table. Notice that it is nearly the same shape as a circle. Now measure the distance (diameter) across your headband (circle).

In my case, if I measure what is approximately the diameter of my headband, I find the length is close to my hat size (seven-and-seven-eighths). How about you? What does that suggest our hat size means?

The students were most impressed by this portion of the reflection. They liked that the hat size number was not some arbitrary value - that it was actually connected to something mathematical. I looked for some history of hat sizes to explain why this value is used instead of circumference, but Google failed me.

Now what fitted hat size should I buy from Lids.com if my head measure is 24 inches? Seven-and-two-thirds doesn't look like it's an option.

In an earlier unit, students struggled with the idea of independent and dependent variables and creating graphs that accurately represent a real-life situation. Because a hat maker does not make all possible diameters, we decided it didn't make sense to connect the dots. Instead, we came up with the graph shown above.

One of the reasons I like this activity is because it does connect with so many other standards, like 6.EE.C.9 and 6.SP.B.4. What do you think? Does this lesson have merit - is it worth saving? If so, how? Please add your thoughts in the comments.

Updated: As much as I loathe Pi Day, this piece on Stormy Kromers (hats made in the Upper Peninsula of Michigan) might make a nice connection.

At what level is his thinking?

Pierre van Hiele

My colleague, Jon Hasenbank, and I have been discussing the van Hiele Levels of Geometric Thinking and what they mean for teaching and learning in mathematics. I am particularly interested in finding videos of people sharing their geometric thinking so that we can apply the Levels and evaluate their thinking. If you are not familiar with the Levels, here's how Pierre van Hiele, the architect of the Levels, described the first three Levels in Developing Geometric Thinking through Activities That Begin with Play [PDF].

In my levels of geometric thinking, the "lowest" is the visual level, which begins with nonverbal thinking. At the visual level of thinking, figures are judged by their appearance. We say, "It's a square. I know that it is on because I see it is." Children might say, "It is a rectangle because it looks like a box."

At the next level, the descriptive level, figures are the bearers of their properties. A figure is no longer judged because "it looks like one" but rather because it has certain properties. For example, an equilateral triangle has such properties as three sides; all sides equal; three equal angles; and symmetry, both about a line and rotational. At this level, language is important for describing shapes. However, at the descriptive level, properties are not yet logically ordered, so a triangle with equal sides is not necessarily one with equal angles.

At the next level, the informal deduction level, properties are logically ordered. They are deduced from one another; one property precedes or follows from another property. Students use properties that they already know to formulate definitions, for example, for squares, rectangles, and equilateral triangles, and use them to justify relationships, such as explaining why all squares are rectangles or why the sum of the angle measures of the angles of any triangle must by 180. 

And now for some practice. Given these descriptions, how would you categorize the thinking of the individual in this video?

Slice a rectangular pyramid: What happens when you slice vertically into a rectangular pyramid? What kind of geometric shape results?

What evidence do you have to support your categorization of his geometric thinking? (Perhaps there are clues in the task's questions.) If more evidence is required, what questions might you ask to get a better sense of his Level of Geometric Thinking?

As always, your thoughts are welcome in the comments - as long as they are civil.

Whose fault is it that you aren't good at math?

Yesterday, Seth Godin posted Good at math on his blog. There was a lot that I agree with in this short piece. For example, the second paragraph begins with:

I'll grant you that it might take a gift to be great at math, but if you're not good at math, it's not because of your genes.

Unfortunately, this is followed up with:

It's because you haven't had a math teacher who cared enough to teach you math. They've probably been teaching you to memorize formulas and to be good at math tests instead.

I am not surprised that Godin employs the "blame the teacher" canard. Our nation loves finding easy explanations to complex problems and, therefore, falls back on the "bad teacher" narrative on a regular basis whenever it comes to problems in education. However, this explanation of why you aren't good at math misses an important point. A point Richard Skemp makes in Relational Understanding and Instrumental Understanding.

I used to think that maths teachers were all teaching the same subject, some doing it better than others. 

I now believe that there are two effectively different subjects being taught under the same name, ‘mathematics’.

Skemp's realization can help us to make an important distinction. It isn't that your teachers didn't care. In fact, probably the problem was that your teachers, like the rest of our society, cared too much - cared too much about you being good at math tests. And this is the crux of the problem (and another place where Godin and I can find some agreement). In the third paragraph, he writes:

Being good at standardized math tests is useless. These tests measure nothing of real value, and they amplify a broken system.

So here is what I wish Godin had written in those two paragraphs (my edits in blue):

I'll grant you that it might take a gift to be great at math, but if you're not good at math, it's no because of your genes. It's because of your experiences. You did not encounter in math class the experiences you needed to be good at math. What you received, because of our broken system's obsession with test scores, were experiences meant to prepare you to be successful in schoolmath - memorization of facts and formulas that can be easily assessed using standardized-tests.

What can we do about this disconnect between math and schoolmath? We can begin by recognizing that being good at standardized math tests is useless. These tests measure nothing of real value, and they amplify a broken system.

What do you wish Godin had written? Because until we can understand the problem, to be able to put it into our own words, it will be nearly impossible to solve it. 

Why do I make these things so hard?

During a Math Institute put on by the Public Education & Business Coalition (PEBC), we were asked to monitor our thinking while working on the Coffee Break problem from (warning, this link includes the answer) Discovery Education. Essentially, we were creating a Metacognitive Memoir that would make our thinking, not just our work, visible for others to see. Being familiar with this form of math writing, I quickly began working on solving the problem.

I started by reading the problem - the entire problem. In the past, there were times when I began working on a problem without understanding it. This lead to "wasting" time on unproductive approaches. I did not want something like that to happen on this problem.

The very first step, "Beginning with a full cup of coffee, drink one-sixth of it," got me thinking about similar fraction problems I have worked on. The second step, narrowed the list of similar problems to those involving mixtures. Fortunately, I kept reading because if I had started solving this problem with those previous approaches in mind I would have gotten off to a "bad" start. This problem was quite different from those I had worked on before.

As I was reading, mental images of coffee cups began coming to mind. At first these were simply generic cups of coffee. But pretty soon I realized that I was going to need a model that could represent the quantities presented in the problem.

In order to make the steps of my thinking visible, I decided to make a series of images, like a story board, that showed the ways I was picturing the different steps. First there was the whole cup of coffee. Then there was the cup filled with five-sixths coffee and one-sixth milk. Next I drank a third of the mixture ...

While I initially took the top third, I quickly realized that the milk would need to be mixed with the coffee. I decided to show this by splitting the cup up vertically. But what was I going to do to show drinking a half of the mixture?

As I was debating how to represent this step, and writing down my thinking about it (I could make a three-dimensional model or split each of the eighteenths in half), the facilitator asked us to stop working on the problem so we could share our results. RESULTS?!?! I felt like I had barely gotten started.

Sure enough, some of the other participants had answered the questions:

• Have you had more milk or more coffee?
• How much of each have you had?

What happened next got me wondering, "Why do I make these things so hard?"

Anyone want to guess what happened (what one of the participants said) and why I made this problem so hard? If you'll put your guesses in the comments, I will update this post with the answers next week.

Who would you want to work with?

We are in the process of making teacher-groups for Family Math Night. The teachers (MTH 221 students) will work together to develop an activity related to specific standards, try out the activity with K-6 students, and reflect on the activity's effectiveness. Throughout the project, teachers use frameworks from the 5 Practices and the Principles to Actions to inform their efforts. This is one of the ways I try to embed the work of teaching into the course.

Because I also want to prepare pre-service teachers to be your future colleagues, I am soliciting your help in identifying norms for collaboration. What are some things you look for in colleagues with whom you choose to work? I am trying to come up with five criteria that the teachers could consider as they evaluate their interactions with their peers.

I have a compulsion to use acronyms, so I made the checklist on the right using some suggestions shared on Twitter. Does this list work for you? If not, how would you adjust it? Please do not be limited by this format as you offer suggestions in the comments.

Thank you in advance for your contributions to the development of these future educators.

Where is the value in play?

This week, Kathy and I presented at the Michigan Council of Teachers of Mathematics Conference. We adapted our previous workshop on games to focus specifically on the Common Core Standards for Mathematical Practice. (Here is a PDF of the session PowerPoint.)

We used the grouping by one of the Common Core authors, William McCallum, to make the Practices more manageable for the participants. Then we concentrated our attention on Standards 7 and 8 (what McCallum refers to as "seeing structure and generalizing"). I shared how some preservice teachers had synthesized this pair into three key elements to look for while doing math:

• Noticing: recognizing patterns by breaking things down and identifying basic structures;
• Building: creating new knowledge by connecting ideas to what is already known; and
• Generalizing: identifying ways to create general methods/formulas

By intentionally narrowing our focus in this way, we hoped to model the importance of highlighting learning opportunities that occur during play.

Participants were given the opportunity to play three games. Two of the games, Race to 100 and Roll a Square, provide opportunities to examine the structure of our place value system and how the structure can be used to create methods for solving double-digit combining and separating problem. We asked participants to explore the games as teachers - keeping in mind scenarios that might be used to highlight Standards 7 and 8.

I provided the following as a model scenario:

While playing Race to 100, I saw Alyssa start on 14 and roll a 10. She ended on 24. What if she rolled 3 more tens in a row? What would the Rekenrek look like at the end of each roll?

While rolling four tens in a row is unlikely, we can use the shared experience of playing the game to provide learners with a chance to notice how our place value structure can be used to build a strategy for adding 10 to a number.

By playing the games before using them with learners, the teachers can be intentional about looking for opportunities to highlight "learnable moments." Then, teachers can use reflection time to talk about scenarios they observed during  game play (during planning, during the lesson, or even imagined). This can be as simple as sharing the scenario and adding one of the questions (from this PDF) associated with the Practice Standard(s) the teacher has decided is the focus of the lesson.

Too often we do not take the time to debrief around games and make explicit some of the mathematical practices that occurred. Is it any wonder that our students respond, "Nothing," when asked by their parents or guardians what they learned in math class today. Let's not leave learning to chance and assume learners will use the skills that will help them to improve their mathematical practice.

Am I Enough?

I am a perfectionist. Somewhere in my past, I internalized the message that my worth was tied up in being perfect. I do not blame the adults in my life for this message. Chances are they were dealing with their own perfectionism. The fact is that I thought I had to do everything perfectly, which has had a negative impact on my life.

In Daring Greatly, Dr. Brené Brown explains that perfectionism is one of the ways we try to protect ourselves from expressing vulnerability. The problem is, it is through vulnerability that we connect with others and access our ability to change and grow. My own struggle with perfectionism has at times left me feeling isolated and kept me from trying new things. As educators, if we are not able to appropriately express our vulnerability, then we risk passing along "gifts" like perfectionism to another generation of learners. (If you have not watched Dr. Brown's TED Talks on Vulnerability and Shame, they are worth your time - especially as you think about setting a classroom culture for the coming school year.)

Part of the problem, is that we are fighting against a culture of perfectionism in education. Here are two examples from this past week. (Be forewarned: one of the problems with being a perfectionist is that I read perfectionism into things where it might not exist. I will readily engage in a conversation in the comments if you think I am wrong about these examples.) First, I read a review of the book, Building a Better Teacher. The reviewer ends his piece with, "Learning on the job just shouldn’t cut it anymore." This seems to suggest that teachers ought to be perfect from day one. Then, I watched Campbell Brown on The Colbert Report. Her new project, Partnership for Educational Justice, is working to help "students fight laws that keep poorly performing teachers in their classrooms." While this sounds "common sense," it ends up creating expectations that teachers be perfect: perfect in their teaching; perfect in their implementation of district plans (even if they are pedagogically unsound); and perfect in student learning (even though they have little control over this aspect of education). Therefore, teachers are expected to be perfect from their first day to the day they retire. As Dr. Brown's research has shown, this push for perfectionism can inhibit teachers' ability to collaborate with peers, connect with students, and be innovative in their teaching. Is this really what we are after in education reform?

The woman who performed our wedding was the campus minister at Western Michigan University when I did my doctoral research. I think it is safe to say that my individual work with her helped me to overcome some of the perfectionistic tendencies that threatened to interfere with me completing my dissertation. She helped me to see that I am not called to be perfect but gracefully imperfect. This is the message I try to pass along to the teachers I work with. For me, graceful imperfection entails: (1) awareness when things go wrong; (2) acceptance so I don't resort to blame; and (3) adjustment so I can grow from the experience.

I still struggle with perfectionism. Now, however, I try to give myself permission to handle the struggle with grace. Hopefully, this post fits into that category.

What are (and aren't) the CCSS?

There is some real confusion about the Common Core State Standards in Mathematics [CCSSM]. For example, check out the #CCSStime Twitter feed from last night. Maybe this will help.

This is NOT an example of a CCSSM:

from Liberty Unyielding

It is a curricular resource selected by a district, school, or teacher to support the development of some objective. That objective might be aligned with the CCSSM, but there is no evidence on the sheet that this is the case.

Here is a curricular resource that claims to be aligned with the CCSSM:

from Create * Teach * Share

The 3.NBT.3 notation in the upper-righthand corner indicates the standard this worksheet is meant to address. It is up to the district, school, or teacher to determine if this resource does indeed meet the standard. Still, this is NOT an example of a CCSSM.

This IS a CCSSM:

You can be sure of its authenticity because it comes from this document, Common Core State Standards for Mathematics. Anything not found in this document (like the curricular resources provided above) are not a part of the CCSSM.

When it comes to development and selection of curricular resources, the CCSSM is quite explicit; it leaves these decisions to the teachers. It does not endorse any set of resources or even a sequence of topics. From the Introduction:

These Standards do not dictate curriculum or teaching methods. For example, just because topic A appears before topic B in the standards for a given grade, it does not necessarily mean that topic A must be taught before topic B. A teacher might prefer to teach topic B before topic A, or might choose to highlight connections by teaching topic A and topic B at the same time. Or, a teacher might prefer to teach a topic of his or her own choosing that leads, as a byproduct, to students reaching the standards for topics A and B.

I am not defending the CCSSM. We can, and should, have a debate about something as important as a national set of mathematics standards. But let us have an informed debate, which starts with actually reading the document.