The Map is Not the Territory

Q: When is a 10 not a 10?

A: When you're seven years old and you've just finished counting all the dimes in the change jar (91).  You've successfully made an array of ten dimes in nine rows.  There's one dime left over.  You count each row of ten: "Ten, twenty...ninety..." and then you look at the lonely little dime and say, "And ten more! 100."

After a hugely frustrating argument conversation about how dimes can be both coins and amounts, your mother ends the lesson (that was, honestly, a little too long but she pushed it a little since it had been going well up until that point) and goes off to wash dishes or sweep the floors or do laundry which is what she does when she needs a little space and time to think something over.

It's at that point when your mother thinks back to something she read about fractions and units in this article from the MAA.  And to a fabulous video lesson by Christopher Danielson and produced by TED-ED on units called One is one...or is it?  She also thinks back to a post by Julie on using Cuisenaire rods for a factoring activity, and, finally, to a book called 1+1=5 that she read about in this Moebius Noodles post.

She thinks and she thinks, for days, actually, and then comes up with a very simple worksheet she titles "Unit Equations!"  The exclamation point is intentional because, really, the world of numbers is an exciting mystery to be unlocked, whether you, the seven year old, know it or not.



















The worksheet poses questions like:

1 carton = 12 _____
2 hands = _____ fingers
4 seasons = _____ year
_____ hour = 60 minutes
1 triangle + 1 square  = _____sides

...and, of course:
10 pennies = 1 _____

Then the questions get a little mysterious.  What does your mother mean by "5 whites = 1 _____"?  That's when your mother pulls out the Cuisenaire rods.  You've not really played with these for over a year, but you 'get it' right away.  Not only that, but you sail through the other five questions easily. 

What you don't know is that your mother was judging for frustration tolerance level during that activity and has decided to see if you can go a little further.  She worked up a little activity on Saturday night which included finding the factors of each factor of 3 up to 15.  The basic instruction is that, no, we're not adding, we're finding multiples.  Each row of blocks must be exactly the same length (thanks Julie!) and each row must be a different color.

This activity, overall, is in the 'just-right challenge' category, but there is still some confusion.  Two light greens (2 x 3) is the same as all whites (6 x 1) is the same as reds (3 x 2) is the same as dark green (6 x 1) which looks blue in the picture. 



















"What does that mean exactly?" you ask. 

Your mother stands the dark green rod (6) on its end and says, "This rod is 6.  There is one rod to make six.  So, that's 6 one time.  The white rods are each 1, and there are six of them. So, that's 1 six times. The funny thing is," and here your mother pauses, "...can I give you a super big word?  It's called the commutative process.  That's just a fancy word that means that no matter what order you multiply or add numbers, the result is the same."

And, as your eyes do not convey complete understanding, your mother continues, "What that means is that even if there are two 3-rods or three 2-rods, they are both the same total length." 

From there, the activity proceeds wonderfully, all the way up to five 3s, and when it is all over, your mother asks hopefully, "So, what did you notice while you were doing this activity?

And the seven year old replies succinctly, in her sweet voice that resembles nothing of the frustrated girl from last week, "That one thing can also be something else with many parts."

YES!!!

Fun, but as my seven year old daughter said "I know my tens times tables," and I think that would be most folks' assessment of the book.  But...

...if you read all the way to the end into the section just past the poems you will find a gold mine!

Multiplication to me has generally always been about memorization.  I've been trying really hard lately to push through my wall of resistance about numbers toward something that approximates actual understanding -- I want my daughter, as well as myself, to understand what multiplication really is.

What I've come to learn is that multiplication can be used in a number of different contexts and that there are some basic models that can help a child experience, visualize and conceptualize the meaning behind the operation. Through some reading and a great post from Let's Play Math on multiplication models I've learned that these models include:

Sets or groupings, which we've done a lot of.  Arrays are ubiquitous.  But the third concept, measurement, is the one I am least comfortable about.  I mean, when I was in third grade I memorized my times tables and that was that.  But I had no earthly idea what they were for or even how to use them outside the context of an equation.  Working through the back pages of Ten Times Better this morning I finally get it.  And, what's even better, my daughter got to pull out the 30' tape measure, run laps in the sun room and do some long jumps all in the name of mathematical inquiry.  Here's are the highlights:

The back of the book holds amazing information about the twenty or so animals from the poems in the preceding pages. The ant is TEN TIMES STRONGER than humans.  If an ant weighed fifty pounds (the weight of a human child) how many pounds could it lift?  My girl counted it up on her fingers and immediately sprang up and ran around the living room trying to lift up all the chairs.  I nixed that idea, but it was such an immediate reaction that it sparked the idea that this needed to be an interactive experience.

Nine-banded armadillos are one foot long.  They are descended from glyptodons who were TEN TIMES LONGER.  I must admit I initially tried to get her to just answer the question using her knowledge of the tens times tables.  I mean, they're easy, right?  But as this activity proceeded I realized very clearly how memorization of facts does not assure an understanding of either how to apply the operation in context and especially not when size and scale are involved.  My kid, at least was not getting it, even with an 'easy' problem of 1x10.  So, I said, how about using measuring tape?  She ran to retrieve the 30' measuring tape and we found the one foot mark.  Then we counted out a total of ten feet.  Wow!  The glyptodon was not huge, but definitely MUCH bigger than it's modern descendent the armadillo.

Some centipedes have 100 legs, but the garden variety has 30.  "With so many legs," the text reads "the centipede really is TEN TIMES FASTER than most insects it catches for food ... If a centipede were as long as a six foot adult is tall, it could run twelve feet in one second.  How far could it run in ten seconds?" Well.  On this rainy day we couldn't go outside, but we measured the sun room at 24' in length.  Five lengths would equal approximately 120'.  Could she run it in ten seconds? (It took her only twelve!)

Laughing, we went back to the living room. Did you know that elephants are TEN TIMES HUNGRIER than you and eat as much in one day as you eat in a month?  If you eat 40 pounds of food every month how much does an elephant eat daily?  I started skip counting by 40 (which is really the same as 4, right?) and she joined in.  400 pounds!?!  Wow.  That one impressed both of us.

Next, the frog. "Most people can jump as far as they are tall," says the text "but a frog can jump TEN TIMES FARTHER."  Before I knew it we had jumped up, taped down a line to jump from, measured the length of the girl and she was off!  It's true!  She could jump the length of her body, and sometimes a bit longer.  Can you imagine TEN times further?   From the back of the house to the front of the house.  Wow.

A baby giraffe is six feet tall when it is born.  The kid jumped up on the couch to measure how tall that was.  And it is TEN TIMES HEAVIER than a huge human baby.  "If that baby weighs eleven pounds at birth...how much might a small baby giraffe weigh?"  By this point, she really had the concept and jumped in to start counting by 11, fast, marking on her fingers...110 pounds!  She ran off to the bathroom scale.  At the age of seven she's about half the weight of a baby giraffe at birth.  Awesome!

The goldfish question is great -- if kept in a small and/or crowded bowl it only gets to be about 2" long.  If allowed more room, TEN TIMES LONGER!  We pull out the measure tape.  Here is 2" let's count that ten times... Again, the numbers are one thing, and 20" isn't that long, but seeing the tape/fish get longer, and longer, and longer is another.  A living number line!

Our last one was the giant squid that can be TEN TIMES LONGER than the tallest basketball players are tall.  "If a basketball player is seven feet tall, how long would a giant squid be?"  70 feet?!?  How long is THAT?!  Our measuring tape was only 30' long.  Our sun room is 24' long.  So that means almost three sun rooms long??  Wow.

Did I mention I really, really love this book?  Scale is such an elusive concept for me, and I'm sure for kids too.  Ten times bigger, longer, faster and smaller is a large enough amount to make an impact, psychologically speaking, on kids who know intuitively that they are small creatures in an adult-sized world. I read somewhere that intelligence isn't the biggest factor in being 'good' at math -- it's actually personal motivation (and, I would add, personal relevance) that motivates someone to engage in mathematical activity.  This book provides motivation in spades.  I think that kids from preschool to middle school could all get something out of physically measuring out 'ten times longer/bigger/faster' to figure out the answer instead of just calculating it.  Just because you've memorized something does not mean you 'know' it.  I am living proof!

p.s. I just found this book.  No one paid me for a review.  But, if asked I'll say it's TEN TIMES BETTER than any other math book I've read in a long, long time!

There are times when I doubt my approach to teaching and learning math with my seven year old daughter.  The approach relies heavily on having faith in a learning process that cycles through input-conversation-incubation-inspiration-output and back around again.

Every time I wonder if there's anything sinking in, or think that we're not doing 'enough' of the 'right' kind of math, I invariably run across jewels like these which put my mind and heart at ease...at least until the next fallow period.

A week ago we were both playing around with a nice Christmas gift of a felt board, colorful felt triangles and some awesome design cards.

















Yesterday I challenged her with some of these designs and watched her approximating and adjusting angles, creating strategies for reproducing the designs, turning and flipping the triangles, and predicting the growth of the triangle pattern in the upper left corner.


Today I found this on the making table, buried under some other projects.  It looks like she was drawing (mostly) isosceles right triangles, y'all.  Awesome! 


































Also of note, the child sat herself down today and, after weeks of conversations about what she might want to do with a pool of money collected from various sources (holidays, allowance, odd jobs, etc.), devised this budget. (Annotated translation below.)


































It says this:

I have $50
Spend $25
Save $25

Cocoa [hand-made stuffed kitty] spending $5 [Meaning she's putting aside that money for things it 'needs'.]
Biscuit [lovey kitty] spending $5  ["And then I'll have $15 to spend on myself Mama!"]
Things I need or want too expensive for me: my own ice skates

List of things Biscuit needs: 4 pairs of socks

Me: "Why four pairs?"
Kid:  "Because he has four legs, silly!"
Me:  "But pairs come in two...does Biscuit have eight legs?"
Kid: "Oh, right!  He'll need two pairs of socks!!"
[Me, thinking to myself: iconic multiplication!!]

Things Cocoa needs
__________________

We do all kinds of math these days -- reading living math books, games, design activities, drawing, noticing things when we go out, engaging in loooong conversations about the fate of her Christmas money and, yes, some workbook stuff.  Taking the long view, I suppose that what I want most for her is that math becomes both useful and personally meaningful for her.  I think today proves we're headed in the right direction.

I'll start right out by saying that I'm pretty sure this was not the right activity at the right time for my 7.5 year old darling girl, but I did learn a lot about where her mathematical thinking is right now, which is always helpful.

In the last one and a half years my inquiry into elementary math education has kept pace with her math learning.  We've discovered so much together and it's been an incredible learning process for both of us.  Lately, though, it seems like the big picture concepts have clicked for me but as I try to move forward myself I end up rushing her.  The following activity is a case in point, but I still think it has merit for some child, somewhere!  Here's how it played out:

Back in November I bought a travel set of attribute blocks.  We haven't done much with them yet, but after looking through the little activity booklet that came with it I found an activity that piqued my interest.  It had to do with figuring out how many blocks of a certain shape (or combination of shapes) you would need to have to get a total number of sides.  It looked vaguely algebraic to me but was presented as a mental math activity.  So, I thought I'd create my own version of the activity on paper to make it a little easier to follow and to visually reinforce the differences between shapes.

This is the first worksheet I made.  In the first example I labeled the triangle with a 1 (meaning one group of three sides) and she had to figure out how many more hexagons made it add up to a total of 15 sides.  The second problem also had one shape already labeled, but in the final two problems I left it to her to figure out how many of both shapes.  You can see little pencil marks around the shapes at the bottom where she counted the sides one by one and then made notes for herself.


































She was not completely happy with this activity (and was in a bad mood, distracted by whether the word 'futzy' was an insult or not).  Grumpy or not I think it really stretched her capacity in a good way, well enough for me to try again.  In the second iteration I asked her to write the total number of sides under each shape which I think really helped.  It was easier for her this time around.  You also might notice that I rephrased the question a little.
























Then the holidays interceded with math learning.  Over that time, though, I did some thinking about how perhaps this kind of activity could be used to reinforce the concepts of multiples and the commutative property.  For example, 4 three-sided shapes (triangles) have the same number of edges as 3 four-sided shapes (squares or rectangles).  I also wanted to continue to stretch her idea of what the equal sign means; not necessarily a result, but a relationship -- various expressions of the same idea.

Here is the third activity.  In it I intentionally grew the numbers from 6 to 12 to 24 to 48. 






































This time her strategy right out of the gates was skip counting whole groups of sides to work toward her answer instead of counting individual edges.  This means to me that somehow in the last three weeks her brain has begun to 'group' with more facility. I think this because, in the same time period, she has also experienced a huge jump in her reading abilities -- from having to sound out familiar words as if they were new every time to simply looking at a word and knowing what it says.  The math concept of 'grouping' and the reading concept of 'chunking' are essentially the same skill -- smaller items grouped into a larger whole. I saw that click into gear today with my daughter as she went to skip counting unbidden.

Anyhow, she moved through this last activity fairly quickly until the last two problems.  After it was finally over she proclaimed, "That was hard!  I hated it!  Forty-eight is such a big number!!"

That proclamation was revealing to me -- at this point in the game she's got facility with multiples of 0,1, 2, 3, 4, 5, 10 and 11.  The larger numbers are still a lot of work in terms of multiplication.  Being able to decompose a number like 48 was just too much at this moment in time. Ultimately, I think it's a call to put on my own brakes and, instead of trying to rush us forward, really dig into the mysteries of number composition and decomposition.  I know numbers are my weak point, so this will be good for me personally as well. 

Epilogue: After drafting this post this afternoon and then leaving to let it sit for a while I ran across the multiplication card game called Snap it Up which I found a month or so ago while at Goodwill (read about moreof my thrifted math here!).  I decided to give it a try and what do you know?  It was fun for both of us!  One interesting observation was that when I said 'what's x times y'  she'd give me a blank look but when I said 'what are two fives...' or 'how many tens make eighty' she totally got it.  I love it when the math stars align for us like this.  It happens a lot, actually, but I am grateful each and every time.