Learning Math by Ear

At first glance, this article about the value of reading aloud to older kids would not seem to connect to math learning. But, to me it does.  Here's the piece that really stood out:

“The first reason to read aloud to older kids is to consider the fact that a child’s reading level doesn’t catch up to his listening level until about the eighth grade,” said Trelease [a Boston-based journalist, who turned his passion for reading aloud to his children into The Read-Aloud Handbook in 1979], referring to a 1984 study performed by Dr. Thomas G. Sticht showing that kids can understand books that are too hard to decode themselves if they are read aloud. “You have to hear it before you can speak it, and you have to speak it before you can read it. Reading at this level happens through the ear.”

Did you catch that? "You have to hear it before you can speak it, and you have to speak it before you can read it."

I made a similar point while working with teachers and teaching artists in Minnesota last week when participants noticed how the math language was woven seamlessly into our dance work. 

This vocabulary development, I said, was initially an attempt to help kids pay closer attention to what they were doing while they created their dance patterns. I noticed that they became much better creators when they had the right words to help them identify their movement choices.  When I started developing the Math in Your Feet program (after about six years of teaching clogging and developing the Jump Patterns tool) I put all those words into a poster that became an ongoing resource in our classroom:

Here's how it all connects to the hearing/speaking/reading continuum.

In my classes at first we just dance, getting a sense of the new movement vocabulary and style of percussive dance in our bodies. I talk us through our dance learning by demonstrating with my own body and also saying the descriptive words out loud during our group warm ups and dance time.  Sometimes I point to the words I've posted while I'm talking. When students look confident with the dancing I ask them to verbally label the specific attributes of their dance patterns, using the Movement Variables chart as a resource. (Not incidentally, attributes are also a 'big idea' in mathematics.)

I add vocabulary words as we go along: yellow is dance, blue is math, and green are where dance and math ideas overlap.   This picture was taken about half-way through our work together.

In addition to being able to parse our patterns, we use tons of other math terminology while we choreograph with our teammates and also in whole group discussions.  This approach allows students to fully grasp the real meaning and application of these ideas which, ultimately, allows them to write and talk confidently about their experiences making math and dance at the same time. Teachers consistently notice an increase of 'math talk' in their classrooms during the time I am in residence. As in, "I couldn't believe how much math vocabulary they were using!"

Math is a language but it's not just about terminology; there is also a need to become fluent in playing with mathematical ideas, noticing patterns and structure, sorting and comparing, and reasoning out and understanding relationships. Similar to the point made in the article, linked above, these math ideas are ideally facilitated by an adult through conversation, play and exploration before bringing it to the page.

How do children learn their native languages? By first hearing and playing with the sounds of words, experimenting with the rules of the language, testing what effect those words have on the world and the people in it, and fairly quickly coming to understand that words are useful in all sorts of situations. It's the same for math learning.  Similar to literacy (not just decoding) a child needs to 'hear' the math first (talk it, do it, play with it, manipulate it, make it, see it) before they can abstract or 'read' the math (notation, symbols).  

Ultimately, what this means is that not only is it important to get a sense of the rhythm and the flow of the math/music/language first, but it is also really helpful to learn things in context.  Words mean something, language is a two-way street, and nothing means anything unless it has a use. You may be able to memorize a definition (or a math fact) but if you never get a chance to see, hear, feel or use a math idea in any meaningful, useful or fun way, it's really just a spectacularly sad example of 'in one ear and out the other'.

Luckily, there are many wonderful resources for creating contexts in which you can help children (or yourself!) learn math 'by ear' through games, puzzles, interesting videos, conversations, living math books and math art.  Here are a few of my favorites:

Moebius Noodles
Let's Play Math
Living Math (especially the list of math readers)
Talking Math with Your Kids
Math Munch
Math Pickle

Edit/Addendum: After I posted this piece I realized that I could have also very easily supported the idea of learning math by ear using examples from the last couple years of homeschooling a wonderful, bright but resistant learner in the ways of math.  I could have also made arguments and brought in examples from my experiences as a traditional musician and dancer (Irish flute and a variety percussive dance styles from Canada, the U.S. and the British Isles) where music and dance is learned and taught as an aural tradition, very literally 'by ear'.  

My Favorite Math Games & Interactives: Online & Off

A friend of mine has a couple kids in kindergarten right now.  He asked if I had any recommendations for "online computer games for arithmetic, patterns, creative math exploration."  Why, yes, I think I do!  Following is an annotated list of some of my favorite interactive online habitats for a wide range of ages, including parents.
 
However, I think I'll start with a list of activities that are not computer based.  I'll be the first to admit that I'm a little biased toward hands-on, not-virtual learning, especially for the primary grades; children's bodies really need to be involved in the  learning process, especially math.

Also, although my kid has some computer skills, she is basically raising herself as a luddite, which is fine with me for now.  She's never been drawn to the computer, even when I specifically show her something I think is fun or cool.  So, many of the games and interactives listed  in the online category, below, are places that I go to get activity ideas to adapt for her off-line lessons or to grow my own math understanding or just for inspiration.

OFFLINE FUN
I don't know where we'd be this year without the endless Shut the Box tournaments of last year.  I attribute at least half her mental addition skills to rolling two dice (which can make up to 12) and finding different ways to make the rolled number.

I attribute the other half of my daughter's mental addition ability to UNO.  It's a classic game that we played speed style all last year.  And, at the end, the winner got to add up all the points she won from her mother opponent who always remembered to chant the UNO Point Tallying Mantra: "Find your tens!"

Peggy Kaye's book Games for Math  has LOTS of easy and inexpensive ideas including Nine Men's Morris which is sort of like a more complex and moveable tic tac toe (strategy, logic).

Variations on "sort and count the change jar" along with "save and spend" from about the age of five means that now, at age seven, the child is a whiz at money math (also known as adding and subtracting 10s, 100s and 1000s).

And, of course, The Game That is Worth 1000 Worksheets from the Let's Play Math Blog -- for the littles all the way through high school.  This fall we used a version that was a combination of sums and differences.


ONLINE GAMES & INTERACTIVES (NOT JUST NUMBERS)
Some of these are for you, lots for them, and some for the whole family (like Scale of the Universe 2 and the triangle interactive!!)

National Library of Virtual Manipulatives: Truly a class act and a challenge for all ages. 

Math Pickle: This link takes you to a large library of interesting videos and ideas organized by grade level.

Kaleidograph:  I first found this toy and played with it online, and then I bought the real thing for Christmas. Both the online and hands-on versions are great but, no surprise, I find more meaning in the actual toy.

From my childhood memory bank, a classic Sesame Street clip with a lovely song: Two Little Girls and Their Little Doll House.  My adult eyes immediately saw one-to-one correspondence but my little girl heart remembered something different.



Scale of the Universe 2: I have no words for this interactive graphic other than just GO.  You can turn off the music if you want.

I've included this link to Planet Seed/Math Puzzles for Kids because I don't want to forget about it.  It was on a long list of bookmarks already.  Puzzle topics include combinatorics, algebraic thinking, topology, probability, etc. 

Games from Freudenthal Institute look great, and they even have a star interactive!

Make your own isometric dot graph paper here.  These and the other dot graph paper you can print out are fun to leave around the house so kids can make designs whenever the mood strikes.

Math cats!  Need I say more?

I personally love Symmetry Artist from Math is Fun and my daughter actually liked playing with it for a while. It's a great way to really experience the differences between line and rotation symmetries and you can print out the designs you make at the end.

And, you really cannot miss this triangle interactive from the Triangulation blog.  Truly.  Run, don't walk.

To close it out, here is a lovely and thought provoking classic Sesame Street animation with music by Phillip Glass called The Geometry of Circles.

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.

A Small Moment of Multiplication

Last school year I made my peace with numbers, but these days I am actually starting to feel a bit of mastery, albeit at a very elementary level.  It's a long story that starts with the seeds of math anxiety that were sown as a five year old trying to count and match bumblebees on a worksheet. Over the years, it turns out that geometry is really my thing, but numbers, for the most part, give me a general and unwelcome sense of dis-ease.  That is, until I became responsible for my daughter's math education.

Over the last year both of us have increased our capacity for mental  math, which basically means we are now able to see numbers essentially as flexible, interesting objects that shrink and expand as needed.

We have developed our addition and subtraction skills mostly through games and other play and narrative approaches.  For first grade we played hours and hours of UNO and many, many rounds of Shut the Box.  We rolled miles of dice.  We counted, earned, saved, and spent mountains of coins.  My mantra was 'find your tens' and for a long time, it seemed, tens were hard for her...until one day they weren't.  Now she's a pro at double and sometimes triple digit mental addition, and subtraction is not far behind.

She and I have different approaches as we work toward an answer, but it's frankly just awe inspiring that I can add two, three or even four 2-digit numbers before reaching for the calculator.  And my kid?  I wish I had had her skills at seven -- I would have probably loved math decades sooner if I had actually been supported in developing the numeracy she already has. 

Anyhow, this fall I tried to bump things up a little with a math version of the card came War using multiple cards to find sums and differences.  After a few weeks of this it's become obvious that addition/subtraction are not a big enough challenge for her any more and so I've been searching and searching for something else.  The kid has some basic understanding of multiplication and division through a lot of conversations and play with halves and doubles but that's not quite enough any more either.  So I wondered if there was a multiplication game out there that wasn't just about memorizing times tables...

...and then Denise at Let's Play Math wrote a fabulous post about multiplication including a section on multiplication models which included a card game she created!  (It's a free pdf download and about half-way into the post.) You can play the game in the form of Concentration, Go Fish or Rummy.  We spent some time sorting out and getting familiar with the different models (sets, measurement and array) and, after we played Go Fish a few times, all of a sudden the light bulb went off in my head about how I can bring those concepts of into our daily lives.

I may be more impressed with myself than need be, but I mean, just listen to me as I walk my daughter through using the juicer this morning -- I actually sound like I understand what I'm talking about!

Me:  Do you see the 4 on the measuring cup?  That means four ounces.  You're going to juice four ounces which is also one-fourth of cup.  When the juice gets to that line, you'll be done.

Juicing commences.

Me: Okay, you're done.

Kid: That's not enough! [As I pour it into a glass that is narrower than the measuring cup.] Oh, wait.  It looks like a lot less in the measuring cup.

Me: Why do you think that is?

Kid: Well, it spreads out more in the cup.

Me: Yep.  Hey, if you're drinking four ounces of juice and that is one-fourth of a cup, how many ounces make one whole cup?

Kid drinks and thinks.

Me: One-fourth is one of four parts.  Each part is four ounces.

Kid: [Done with juice and looking at her fingers.  She pulls up four fingers and starts skip counting...] Four, eight, twelve, sixteen!!

Me: Sixteen what?

Kid: Ounces.

Me: Excellent!

I feel successful just being able to have that kind of conversation.  And really, it's these kind of small steps that ultimately made last year's math work so successful.  So, score one for a small moment of multiplication!  We're on our way -- and I can't wait to get the Multiplication Models poster I ordered off of Amazon from Maria Droujkova and Natural Math.  It'll be great to have it around us as we find more small moments just like this one.

Blog Carnival over at Let's Play Math!

Hey y'all, I've been included in the "Math Teachers at Play" blog carnival over at Let's Play Math!

This particular carnival features "a smorgasbord of ideas for learning, teaching, and playing around with math from preschool to pre-college" and is illustrated with some great examples of living math books.

I'm pretty excited to be included in my first math blog carnival but least you think I have my math act all together, let me assure you this is one big adventure I'm on.  My blog has been, and continues to be, a vehicle for me to figure out for myself what math is.  I've got a pretty firm grasp on the math we use in my program Math in Your Feet, but I'm a big picture person and I've always wanted to find a way to put our dance work firmly in context on the mathematics map.   I am also perpetually curious about how people teach math at an elementary level.

Luckily I've got a six year old daughter along for the ride.  I use her interests to frame my inquiry and, by working at the pace of a first grader, I am much less intimidated about re-learning math than I might be otherwise.  

Working with my daughter has actually been, to use a new math term I've recently learned, a kind of recursive, looping process.  She shows me, through her drawings, and conversations, and number play, how she is thinking and then I go find great resources like Let's Play Math, or Living Math, or Natural Math, or Love2Learn2Day, and see if I can figure out how to support and deepen her questioning.

In the process of finding a new math game or book for the kid I also find some new bit of information for myself about a math topic or an instructional approach or theory.  I spend part of my days with kid math and my nights reading everything I can get my hands on, from research studies about cognition in mathematics to books about archetypal relationships between mathematics, arts, science and nature.  Then I'll go to bed only to wake up to hear what my daughter's got going around her head, and the process cycles around again. 

I hope you will take some time to check out the Math Teachers at Play blog carnival!  If you are interested in doing and making more math with your own children or students it's really just as simple as finding an interesting math-y story book or game and going on from there.  My biggest lesson in the last year or so is that I don't have to know everything, I just need to keep asking questions!  That's really all there is to it.  Have fun!