Half A Cake

"Hey, tomorrow's your half birthday!  I know, let's make a cake to celebrate!"

Turns out the recipe called for 1 1/2 cups of flour.  Since our cake was ingredient free due to multiple food allergies, we used even parts of two gluten free flours.

"Okay so we need one and one half cups of flour.  How many halves in a whole?"

"Two!"

"Okay, so here is the 1/2 cup measuring cup.  One scoop of the buckwheat, one of the rice... Now, we need 1/2 cup more flour, using both our flours.  What's half of 1/2?"

"One-fourth!"

"How'd you know that?"

"Oh, I just knew..."
..............
Later in the morning, at the library.

"Hey look Mama!  The [really big, red tiled] square is made up of smaller squares!"

"Very cool.  How many small squares make up the big square?"

"Three, six...nine!"

"So, you wanna hear something interesting?  When you hear someone call a number 'square' that's what it means.  Look, nine squares literally make a 'square' number!  Isn't that cool?"

"Mmmm hmmm..."
.............
After lunch.

"Hey, it's time for cake!"

"YAY!!"

"But we can only eat half.  How would you cut it in half?"

[Girl gestures a line from the middle of one side to the middle of the opposite/parallel side.]

"Well, that's one way.  How else could you do it?"

[Girl pauses and then gestures diagonally from corner to corner.]

"Excellent!  But I have one more idea.  How many squares made up the big square at the library today?"

"Nine."

"So, I"m going to divide the cake into nine parts.  Here's one for you, and one for me. We'll have more later, after dinner."
...........
After dinner, and after one more piece each for the child and myself.  (Papa doesn't like chocolate, so no cake for him!)

"Mama, can I have a little more cake?"

"Hmmm, maybe.  We're only going to eat half of the cake -- let's see if there's any left....  How many pieces did we have to start?"

"Nine."

"What's half of nine?"

"Four and a half."

"So, together you and I have eaten four pieces.  There's half a piece left and you can have it!"

Ladies and gentlemen, I present to you: half a cake!

Chicken or the Egg? A Math Integration Tale

I recently had the opportunity to have my teaching work critiqued by a group of colleagues.  They viewed a ten-minute video I produced which illustrated what success looks like in my classroom. The feedback I received was all at once enthusiastic, thought provoking and puzzling.   

I teach elementary students the elements of percussive dance and then, within a structured framework, give them the freedom to create their own percussive patterns.  Along the way we use and talk about a lot of math which both describes their patterns and informs their creative choices.  It seems straightforward to me, so I think that is why I was flummoxed by a question they all had:

“When your students are choreographing their percussive dance patterns, how much of that activity is about their math understanding?”

I can answer that question.  The answer is, “All of it.”  I don’t see a separation between the two.   In fact, I think the dance and the math are essentially the same activity.

Here is an example:  a video of some traditional Irish figure dancing with accompanying percussive footwork.  You only have to watch a minute of the dancing to notice it is full of geometry and symmetry and all sorts of other wonderful kinds of math:

The shifting, curving patterns move through space and time while undergoing symmetrical transformations.  The dance choreography explores permutations and combinations of moves and steps by arranging and rearranging dancers at a dizzying rate in time to the music.  The footwork traces invisible maps on the floor.  The math in the percussive footwork is a reiteration of the figure dancing, but on a smaller scale with more specific and precise patterns.  Unsurprisingly, precision is a hallmark of mathematics which has, by a popular meme, been called ‘the science of patterns’.  

All this is well and good, but what my colleagues really wanted were more specifics about my evaluative criteria.  How exactly do I gauge my students --within the medium of percussive dance or with regard to the math?  Again my, possibly controversial, answer:  Both.  

And a question back: Why do we think of them as separate activities?  I think part of the issue might have a lot to do with how we, on the whole, perceive mathematical activity.

Generally conceived, dance is a three-dimensional, kinesthetic endeavor.  Math is rote memorization of algorithms and concepts and inhabits a two-dimensional symbolic realm.  Everything we’ve learned in school bears this out, except that it’s really not true!  When I started to really investigate what it means to do math I found that it’s completely different than what I did in school when I was a kid (and you too, probably).   

And, as I dug deeper, I also realized I had been thinking mathematically all my life – I just never recognized it as a mathematical activity.

One of the things I’ve come to realize is that, really, people who do math don’t spend a lot of time plugging numbers into memorized algorithms.  Instead,they formulate and/or approach questions that don’t have immediate solutions.  They spend time thinking, talk to others, sketch out ideas on napkins (or whatever), and build models.  And then, when they think they’ve got something that resembles a solution, that’s when they start writing it down.  The notation is the end result of a process of questions, trial and error, and conversations.  Sounds a lot like what we do in Math in Your Feet, actually.  Take a look:

When I first started wondering about whether or not there was math in the dancing I did with students, I knew I needed an interpreter, someone who really understood math and how it was taught to children. I was lucky to be connected with Jane Cooney, a classroom teacher with deep experience and love for teaching math.  Our collaboration in creating Math in Your Feet included long discussions about the best ways to retain the integrity of both content areas.  We weren’t going to make up the dance to fit the math and I wasn’t going to make up the math to fit the dance.

We didn’t and I haven’t.  There was no need. There is enough overlap between the two that, if you hit it right, you often can’t tell where one starts and the other ends.  However, I have consciously created specific lessons to identify and learn the math that we’re going to use in our dancing.  Not only is math a tool we need to understand in order to use it properly, but I think it’s also important to know exactly how math is involved in our physical and creative work.     

Like the old chicken/egg conundrum, it really doesn’t matter which one comes first because they’re both part of the same process.   And that is why, when I watch my students share their work throughout the week, I can see clearly if they have both the dance and the math and to what degree.  But that’s another story!

[This story originally posted in the Teaching Artist Journal's ALT/space, 11/28/12]

Morning Math

I'm really enjoying math in the morning.

There's something fresh and new and hopeful about mornings lately and, even though I'm not doing anything ground breaking, I'm really enjoying how connected everything seems to be these days, mathematically speaking. 

In mid-October I mapped out a basic math plan when it was clear the girl (now 7.5) needed and wanted more and different kinds of math challenges.  I decided to call the plan 'algebra' because, from what I've read, algebra combines a number of skills and concepts that we have to start learning anyhow at the primary level.  And, because the girl has often balked when I introduce new math stuff, calling it algebra motivates her to give it a try; 'algebra' is a big kid skill, and she really, really wants to be big.  

So, in the morning it's been fun to open my math folders and give my lovely child her choice of math activities.

Solve for x (conceptualizing equality and sameness, sums and differences) or math card games (3 digit mental sums and differences)?

Christmas themed beginner Sudoku puzzles or Factor Dominoes?

Growing patterns or Bean Soup (fractions, multiplication, division)?



















 "What's division, Mama?"

"You know, like when you wanted to see if you were halfway through your reader.  There were 96 pages and you figured out in your head that half of 90 was 45 and then you..."

"Oh, yeah."

Later that morning at the science museum she built this:

"Hey Mama, look!  I got the water to cover the whole area!"

"Cool!  That's an example of division, too.  The water is being distributed evenly across the table."

So, here I am, trying to connect our morning math time to the bigger math picture that I'm constructing in my brain.  Like algebra, as I've mentioned.  What is algebra???  I did some algebra when I was in high school, but failed literally and horribly (although I aced geometry).  But why let that stop me?  After some research I decided that the concepts of balance and sameness, solving for unknown quantities, and growing patterns were all pretty darn interesting and relevant whatever we called it and away we went.

We began by building and analyzing growing patterns from some pattern starters I found.  Here's one she came up with on her own:

And, when we do the occasional worksheet (the one below is skip counting/multiplication) I look for ways to extend the activity forward, even if I'm not completely sure I'm right.  The sheet below was for figuring out numbers of feathers on different numbers of hats, legs on rabbits, and petals on flowers.  She got the skip counting patterns easily, so I described the data another way by saying:

"The number of hats you have is multiplied by three feathers per hat which gives you the total number of feathers," while writing h x 3 = f.  The next two examples she talked herself through the whole thing and wrote it down.   But maybe it would have been better to say, "The total number of feathers you have is equal to the total number of hats multiplied by 3"??  Writing all this out has me thinking of another way I could have done that, but I think sometimes its okay for me to muddle through stuff like this.

And, if skip counting is coming easily, why not challenge her to fill in the chart backwards instead? Ooooh, that was cause for consternation, but she had been saying "This is easy, this is easy, " all morning.  I know for sure she needs more experience with subtraction, I know being able to invert a procedure or concept is an important skill, and I also know that if something is too challenging she'll just give up.  So, here was an opportunity for deepening her experience with a skip counting chart and providing just the right amount of challenge at the same time. 

Filling in the charts backwards really made her think.  That's what she wanted, right?

Anyhow, it's nice to be entering a stage with her where we can sit down and 'do math' outright instead of trying to leave it around the house for her to discover, although I'm keeping that strategy in my kip for now!  More than anything, it's been lovely doing math, in the morning, with the winter sunlight streaming in, with a child who is finally (finally!) allowing herself to be interested and excited about exploring this mysterious and wonderful subject.

Shhhh...Sneak Peek!

My new website is up in its temporary home waiting to have the Math in Your Feet address pointed to it.  In the mean time, I thought you might like taking a look.  It's got a ton of new features including an interactive News page and the brand new Math by Design section.  But the best thing of all is...

...a new two-minute Math in Your Feet video!!

I'm about to swoon, truly.  I've had the footage for a year or more, but only recently found the golden key that unlocked its riches. 

While you're there and if you're so inclined, I'd love to hear any feedback or comments you might have.  You can leave your thoughts here or in the comments of the News page.

You can find the site (for a few days) HERE.  

New Math Game: Factor Dominoes!

Lately I've been looking for different ways for my seven year old and I to conceptualize multiplication. As has happened many times before on our math journey, this graphic showed up at just the right time (albeit somewhat circuitously through the excellent influence of the Math Munch blog).

 

 

My favorite thing about it is that it's not about numerals; when I look at factoring trees I can make some surface sense of them, but my mind goes numb pretty quickly. In this visualization, however, there is an incredible connection to shapes and grouping. I find this visual especially well-suited for kids in general and at least this adult specifically.

Last night I printed out the graphic and left it advantageously on the kitchen counter. I thought maybe my kid might be interested but was truly surprised by her reaction when she found it this morning.  It is probably the first piece of math my daughter has ever admitted she was excited to know more about, which is saying a lot.

She wondered what it was about so we looked it over together.  At first it was basically 'count the dots' and notice that each configuration was one more dot than the one before. Then, in the same way we tackled the 100's chart last winter, we started looking around and noticing things: The ring of seven dots on the far right column has multiples of seven underneath it.  The 6 shape shows up two more times on a descending diagonal. It's fun just to look and talk about what you see.

It's the geometry of the design that really shows the relationships between numbers. And, even though this was not meant to be a multiplication chart, it's probably the best one I've ever seen.

All our talking and looking got my mind spinning. What if...what if I made little playing cards out of each factorized number? What kind of game would it be? 

I was about halfway through constructing the cards when my big AHA! moment hit. As I made and sorted them one by one it became completely clear to me that the integers 1 through 7 formed shapes that were echoed in the other factorizations.  As an attempt to organize my growing pile of cards I laid out a top row of 1 through 7.  But where to put the other cards? For example, 5 is a pentagon made out of single dots and 10 is a pentagon group of two dot groupings. Where does it belong?  The 2's column or the 5's column? This kind of question is at the heart of the new game.

Here's how my daughter decided to sort them in a 'get acquainted' activity before we started playing:

As we went along I refined the language she needed to help her make her choices. Was she going to place a particular card based on its large grouping (outer shape) or the smaller groups? As you can see above, there's a 5 shape of 3s in the 3 column, because the smaller group is a match to that number. But, every other 5 shape is in the 5's column. She's also got a 7 shape in the 3's column for the same reason -- the smaller grouping matched and, ultimately, the whole 3's column is consistent on that criterion.

For some comparison, here is how I sorted the cards, earlier in the day. I was trying to match to the category of 'outer shape':

I'm not sure I got it the way I wanted it, but no worries.  There is probably no one right way to sort these cards and the activity in itself makes for some really interesting thinking and conversation.

After she familiarized herself with the cards we started in on the new game which I'm calling Factor Dominoes (with a side of Scrabble). The title alone should give you clues as to the game's aesthetic and procedure, but here's how to play:

Split the deck equally between two players. Player 1 puts down the opening card. Player 2 tries to find a match. If Player 2 has no match the card is put aside face up for future use and play returns to Player 1. You can find a match either by outer grouping/shape (triangle, square, pentagon, weird six shape and seven ring) or by similarity between the small dot groupings. In our game we also matched 'echoes' -- small groupings that are the same shape as another number's outer shape.

For example, in the picture below the first card is a 5 shape with small groups of 2.  The 6 shape next to it works because even though it's a different shape it also is comprised of 2s. And, the card directly below the first card also works because the smaller groupings of 3 match the 5 shape of the larger grouping. Make sense? 

Here's another example: The top line of matches have the 3 shape in common. The bottom row connects to the top with small groupings of 4.

And, here's a picture of a couple more interesting matches.  See if you can figure out our reasoning on this section of the game:

Play the game until there are no more cards. This is a cooperative/conversational game but feel free to give it a point structure if you like. You can also make the game bigger and more complex for older students -- just cut out more factors and make more cards! That's what I'm going to do for our next round of play.

Here is our completed first game:

 

Based the exponential growth of my personal understanding of primes and factors, gained in just one short day, I am firmly convinced that a wide range of ages, experiences and abilities can get something of value out of this game. 

My seven year old was perfectly challenged as we focused on groupings, but what if you added the prime numbers beyond 7 into the mix? How would that deepen or change things? What about adding exponents as a match category? What if you figured the value of each card and matched them in sequences (like {25, 26, 27, 28...} or {4, 8, 12, 16...} or even a sequence of primes, in order)?

If you do play this game PLEASE let me know how it went and what other ideas you have for it.  And, please do consider joining us on the Math in Your Feet Facebook page. We're having a good time over there!

____________________

Malke Rosenfeld delights in creating rich environments in which children and their adults can explore, make, play, and talk math based on their own questions and inclinations. Her upcoming book, Math on the Move: Engaging Students in Whole Body Learning, will be published by Heinemann in Fall 2016.