Saturday Resource

I'm catching up after a long weekend! Also, I'm going to scale back just a bit for my own sake--but I still have MANY blog posts planned (more than 31, really), and I will keep posting, but perhaps not just every day.


Today's resource is just for parents. It comes to us from the Council of Great City Schools. Much like last Saturday's National PTA resource, it gives you a nice grade-level breakdown of what your kiddo will be learning that year, and suggestions for how to engage with teachers.

What I really like about these guides is that they're longer than just 2 pages (these are 6), and each picks out a few key topics, and shows you what content came before, and what content will come after. This way, you get a sense for the flow and the continuum of learning, from grade to grade.

Here's an example from the third grade guide:

These guides also give additional resources at the end, often tailored to the content relevant to your kid's specific grade.

Check out all the various grade level descriptions here.

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 our facebook page, at 9p ET!


Friday Math Task

Happy Friday! Today's Friday Math Task is a fun one. It draws on the children's book "The Very Hungry Caterpillar" by Eric Carle. It's more of a whole-class exercise, so what I'm sharing with you today are the directives for a teacher to facilitate this activity. Here are the directives:

Materials: "The Very Hungry Caterpillar" by Eric Carle.

Students work individually or in pairs. Each student or pair needs:
  • Three ten-frames for each student or pair of students (see PDF for black line master)
  • 30 counters or unifix cubes per pair of students
  • One small dry-erase board and dry-erase maker per pair of students
Read the book to the class and asks, “How many things do you think the caterpillar ate in this story?” The students take a minute to share their estimate with a partner.

Next, reads The Very Hungry Caterpillar again. After each page, pause so that the students can add counters or unifix cubes to the ten-frame to represent the number of things the caterpillar ate, and then write an equation on the dry-erase board connecting addition to the number of counters used. After each ten-frame is filled in the students move to the next one.

If the students are working in pairs, one student can add the counters/unifix cubes to the ten-frame while the other student writes the equation. By the end of the story, there should be a total of 25 food items eaten and 1 leaf eaten. (The students can decide as a class whether to count the leaf as a food). There will be two ten-frames completed with 5 or 6 counters/unifix cubes on the third ten-frame.

If students come up with different, but correct, equations, then discuss the different equations and ask students, "Can all of these be correct?"

Sunday I'll post a debrief of this task. If you've got a copy of "The Very Hungry Caterpillar," feel free to try this task on your own before I share out on Sunday.

And have a lovely 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! This week's Tuesday night math chat will be on our facebook page, at 9p ET!


What's really wrong with the Common Core: Phds speak out.

I'm going to shut up today and let other people do the talking. People who out-rank me in education and experience when it comes to math education, it's history, and doing it right.

Wu and Frenkel (two super stars in math ed from Berkley) wrote for HuffPo about what is really wrong with the Common Core: poor support for implementation. THIS is what needs to be understood and addressed!!

Read the full text here.

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 our facebook page, at 9p ET!


Where did the Common Core come from anyway?

One of the biggest misconceptions about the Common Core State Standards are where they came from. Depending on who you're listening to, you might be told that it's a federal initiative, or that it's the love child of ed reformers and the privatizaters of education.

I was going to write out a whole long description of where the CCSS came from, but you know what? It's all on the Common Core website, under resources.

They've put together a nice FAQ that addresses almost every misconception I've read or heard.

They also have a whole page dedicated to the process by which the standards were written, vetted, and adopted.

And my personal favorite, a Myth vs Fact page, combating a good deal of the misinformation out there. Think of it as a special snopes edition of the CCSS.

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 our facebook page, at 9p ET!


Standard for Mathematical Practice #1: Make Sense of Problems and Persevere in Solving Them.

Both my grandmother and my mother are excellent cooks. My grandmother is a classic 1950s chef--she has a huge repertoire that includes foods like meatloaf, chocolate cake, and apple sauce--all from scratch. My mother is a classic New York Italian cook. Just thinking of her stuffed shells or artichokes makes me salivate--I'm literally doing it now as I type.

The most interesting (and key) difference between my grandmother and my mother is not their cuisine, but rater the way that they cook. While my grandmother lives and dies by her recipes, my mother cooks by feel and taste. My grandmother has all her recipes written out and she follows them step by step, ever faithful, never deviating. My mother doesn't have anything written down, and if you ask her how to cook her artichokes, it's not guaranteed that you'll end up with the same results, because she tweaks and adjusts as she goes. This is why replicating the tender leaves that melt like butter in your mouth is only possible if you've cooked them right alongside her, to learn by doing with her.

Both my grandmother and my mother are good cooks, and can perform. But the key difference in the way that they cook is that my mother is much more adaptable. She can change course mid-meal to bring the sauce back to the way she wants it to be. My grandmother, on the other hand, must rely on a specific set of instructions, and if things go awry, she either needs to scrap it all together and start over, or live with a sub-par outcome.

I share this story of two cooks because it has real relevance to math education. Many of us were taught that math is a set of rules and steps. If we just follow the steps and rules, we're guaranteed to get the Right Answer™. However, as I discussed yesterday, there has been a movement in math education to give kids an opportunity to engage with mathematics, not just memorize its rules and properties. The way that we have been taught is much like my grandmother's cooking: follow the recipe and don't deviate. The way that math is being taught now is akin to my mother's cooking--learn the way that cooking works, and you'll understand why you should lower your burner if your meatballs are burning on the outside and still raw on the inside.

From the outside, this way of teaching math often seems needlessly foolish and complicated. It seems to take longer and be more confusing. But often those criticisms come from folks who have already decided in their minds that, as a matter of principle, math is confusing. I can tell you from being in a classroom, that if you don't have that already decided, engaging with math can be a fun exploration ... an adventure!

Standard for Mathematical Practice #1 gets to the heart of this issue. The standard states that students who are proficient with mathematics will make sense of problems and persevere in solving them. The full description is below.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
This standard describes a student who is proficient with mathematics--it is a goal for students to grow into as they develop and learn. So if your 3rd grader isn't exhibiting ALL these traits, take a deep breath, let the stress melt from your shoulders, and then let the breath out. It's ok.

Let's think a little about the task I posted on Friday. (For a refresher, check out Sunday's debrief.) It was meant to address a 3rd grade content standard. And within that task, I can see a number of elements from the above paragraph:
  • looking for an entry point (a student would need to figure out how to divide 18 by 3, in both contexts)
  • analyze givens (the two contexts), analyze relationships (the questions in the task that ask them about similarities and differences)
  • younger students might rely on concrete objects (the multi-link cubes!) to help conceptualize and solve a problem
Well look at that! We found three ways in which Friday's task would help a student to engage with math in such a way that they are exhibiting elements of SMP #1! Here's a little secret for you: that math task was written with SMP #1 in mind, meaning that this is a grade-level appropriate way for a 3rd grader to be engaging with math in a more analytic, proficient way.

Did you notice that with that task, we weren't asking a student to just memorize division facts? Rather, we were asking the student to play with division, see it in different contexts, make observations, and learn by doing. I don't know about you, but this is definitely a shift from the way I learned division. But I see, regularly in the classroom, how powerful that shift can be.

Suggestion: Take a look at some of the math work your child is doing. Do you see evidence of SMP #1? It doesn't need to be embedded in every single thing they do ... but it does need to be present. Feel free to share back ways that you see SMP #1 playing out in your kiddo's math learning.

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!


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!