The Map is Not the Territory

I was going through the Common Core State Standards for Mathematics the other day, to finalize a document that illustrates the connections between the CCSS and my program Math in Your Feet.  I was really focusing on fourth and fifth grade standards, but while I was there I peeked in on the first grade standards as well.  I was curious.

I've got a first grader of my own, and I shepherd her math learning.  I watch her figure out how numbers work and relate to each other and I observe her make observations and formulate questions while experimenting with shapes and other manipulatives.  So, I really liked looking at the CCSS to put all that work into perspective.  The content standards were all pretty straightforward but I immediately started trying to figure out how I could really know if those words meant things my kid knows.   What does her understanding look like in action? 

Luckily for me, it turns out that a great way to assess a lot of what a first grader knows and can do, mathematically speaking, is to look inside my purse!

We were having a late lunch/early dinner at our local co-op.  The plan was to eat a nice balanced meal and then hit the bulk bins for chocolate coconut candies after that.  I was digging around in my purse to make sure I had enough money to get the promised after-dinner treats.

Digging...digging...and finally finding a handful of coins.  That's when inspiration struck.

"Hey!  Let's figure out if I have enough to buy dessert!"

Unfortunately my camera was at home with battery charging, but the following photos are a faithful re-creation of the independent thinking that followed.

Showing no hesitation she went for the quarters first.  Lined them up, counting by 25, dollar by dollar.  Got the dimes in line counting by ten, and finished out that dollar with nickles (by 5's) and pennies (by 1's).  Added it all up in her head, no fingers (you go, girl!).  The grand total? $5.07.

We made stacks from tallest to shortest, quarters, pennies, dimes and nickles.  And then I remembered attributes, and we compared each coin and ordered them thick to thin.  Turns out the phrase 'one thin dime' could also almost be 'one thin penny' by a hair.  You'll also notice that putting them thickest to thinnest lines them up biggest coin to smallest coin.  Hmmm...I wonder how that happened?
















What did all this dinnertime inquiry show me about my daughter's math skills in relation to the CCSS? 

First, it's important to remember that using money is just one of many ways to represent numbers and understanding of numerical relationships.  As an instructional tool, I think that using money is one of the best ways to develop numeracy. You're working with a whole unit of 100, for one thing, and anything less is a fraction of the whole.  Also, the random sizes -- a dime is worth more but smaller than a nickel -- is a fabulous conceptual challenge, and think of all the different kinds of skip counting!  Plus, money amounts are always changing depending on what you save, spend, or find on the ground, so there's always something new to figure out.

So, she can represent her numerical understanding through money and mental calculations, but although she likes to write out simple equations, we haven't done much with adding larger numbers on paper.  After reading Peggy Kaye and Constance Kamii (which I've written about here and here), I'm fine with it that way for now. 

As for the standards, most of 'how much money in the purse' challenge was related to numeracy skills but, overall, the activity showed me that my daughter is gaining mastery in multiple content ares in three of the four content domains:

Operations & Algebraic Thinking: "Developing understanding of addition, subtraction, and strategies for addition and subtraction within 20."
Numbers & Operations in Base Ten: "Developing understanding of whole number relationships and place value, including grouping in tens and ones."
Geometry: "Reasoning about attributes of, and composing and decomposing geometric shapes."

The other half of the CCSS for math is the Mathematical Practices section. The change in my purse also illustrated these goals in action at the first grade level:

"...adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy)."

In this case the mathematics was most certainly 'sensible, useful and worthwhile'!  Although the change in my purse did not asses everything in the practices section, of course, it was helpful to have a spontaneous moment to see up close not just what she knows but how she goes about using what she knows.  On the whole, I feel confident things are going in the right direction.

And, in case you're wondering, the chocolate coconut clusters were delicious!

Okay, here's my special challenge (and I take comfort in knowing that out there are other people who understand exactly what I am about to say).  I am an artist.  I am a teacher.  I am a teaching artist, curriculum developer, website designer, market research specialist, performer, agent, writer, editor, math student, calf wrangler...I am going insane!  Well, not really, but I do feel like I exist in multiple worlds simultaneously, sometimes to the point where I can completely contradict myself and agree with myself in the same moment!

Today I was all over those Common Core State Standards I wrote about yesterday.  I just spent the better part of the afternoon working with the new language.  I'm not making anything new up, just translating what I do, what I've always done, into the language of the most current stated education goals.  I've also been working on a grant project for the second pilot stage of Young Audiences, Inc. Signature Core Services program, which has meant working with new forms and terminology.

Along parallel lines, I have been spending the better part of my after-the-kid's-in-bed time systematically reading all my back issues of the Teaching Artist Journal (I'm into year five at this point).  I started this particular project in December because I figured that, as a recently appointed Associate Editor for this publication, I should probably have a working knowledge of what kinds of articles, topics, research etc. have been published in the past before I move forward with my new position.

Re-reading TAJ's has actually been really interesting and inspiring, and much less an exercise than my husband thought it might be.  It's been interesting because the process of reading them one after another is bringing me the big picture I never had getting each new TAJ one at a time, three months apart.   I'm learning a lot!  And, it's been inspiring, because I've been running into really great thoughts like this one (below), which balance out all my standards-based wonkyness of late.  So, without further ado, here's my standards antidote for today:

I just had a great introduction to the Common Core State Standards (CCSS) for Mathematics!  I have to admit, there's always at least a little trepidation in me when I look at a new set of standards.  I started as a Teaching Artist in North Carolina, then expanded to South Carolina.  Later I moved to Indiana and also travelled to Kentucky on a regular basis.  My programs remain the same no matter where I am, and it's been a bit of a bear to 'translate' what I do into the different languages of each individual state's standards.  But boy am I in luck now!  Now we have the CCSS which are being adopted by most of the fifty states; I am happy about possibly never having to 'translate' my program again, but I'm also really excited about the fact that these new standards are as much about habits of mind and real learning as they are about tiny pieces of specific content. 

I started hearing about the Common Core State Standards back in the summer, I think.  I went to the website and couldn't really figure out what they were about; even though I've worked with standards in four different states, as a Teaching Artist it's still really not my language.  Fortunately, I have an ace up my sleeve in the form of Math in Your Feet's co-creator, educator extraordinaire Jane Cooney.  This last week I was met up with her, and another fantastic math educator, Indiana University doctoral student Lauren Rapaki.  Together, they drew me a marvellous picture.  Here is what I understand so far...

It turns out that there are two strands to the CCSS for math.  One is a content standard -- I've taken a look with Math in Your Feet in mind, and there's still enough there to be relevant to the specifics of what third, fourth and fifth graders need to be learning.  What is exciting to me is the other CCSS for Mathematics standard, Standards for Mathematical Practice:

"The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students.  These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy)."

When we first developed Math in Your Feet, Jane went straight to the NCTM principles (mentioned in the above quote) as a tool for finding the big picture connections between percussive dance and elementary math topics.  Problem solving, reasoning and proof, connections, etc. have always been at the core of what this program is about but, somehow, the process of parsing it all down into state standards and indicators seemed to dilute the importance of the higher order thinking that happens when kids make percussive foot patterns and learn math at the same time.  The reason the CCSS for Mathematics look so promising to me is that the Mathematical Practices standard is at least half of what kids will be required to do in their math classes; my program is a great match for this kind of inquiry and approach. 

Here's the list of specific mathematical practices outlined in the CCSS, all of which describe the core work of Math in Your Feet:

Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique others' reasoning.

Model with math.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.