Part to Whole

A Very Shy Cheetah of Zero Age

The kid told me she was making "a very shy cheetah of zero age."

"Why zero age?"

"Because she's not yet made...oh, and this kitty is named Island [pronounced Iz-land] and my next little kitty I'm going to call Infinity."

Seeing Numbers

I've been thinking on and off about ways to help my seven year old daughter come to know, appreciate, and maybe even love numbers.  Geometry is easy for both of us because it's visual.  But because my math schooling was of the memorize and drill variety I have really had to stretch myself in order to make my daughter's early experiences with numbers more meaningful than mine were.  As it turns out, I'm learning a lot in the process and, as a bonus, I am finding that numbers are quickly becoming an unexpected but welcome gang of amusing little friends. 

Since we're working on conceptualizing multiplication and division these days through various explorations, I thought it might be fun to make 'number cards' to visualize one number at a time.  Using coins (and clarifying their intent as reconciled in my recent post on units: When is a 10 Not a 10?) we looked at all the different ways to split 25 and found lots of remainders.  I suggested 60 after we read that base 60 was popular early on because it can be divided in so many ways. 

As you can see, the results are less than inspiring and it's not clear to me that the child really learned anything more from making the cards in addition to the array work with the coins.

These cards are still taped up hopefully on the wall, but I had basically abandoned the project...until I saw this fantastic post  from an international school in France that re-imagined factor trees in the most awesome way.  Isn't this the most lovely factor tree you've ever seen?  And there are more -- all as beautiful as this one.  Their project in France has inspired us here at home and has become a perfect and flexible solution for our nascent number wall idea:

I showed the girl the post that inspired me and she was entranced and eager to get started.  She is doing well with hands-on and mental multiplication and division, but factoring as an activity is new to both of us and so I decided 20 would be a nice easy number to start with. 

We took out the Cuisenaire rods and she found all the factors of 20 herself.  She double checked every rod from 1 to 10 and noticed a lot during the process, skip counting as she went.  "If I can use fours, I can use twos, too!"  "Threes...no that makes twenty one...."  

And then we wrote them out as equations and sketched a preliminary tree.  And here is where this activity became as important for me as it was for her.  I do not recall in all of my fourteen years of school mathematics (including those awful matching bumblebees in kindergarten, my tenth grade algebra D, my tenth grade geometry A, and a deliriously confusing semester of college algebra) ever learning about factors.  Ever.  So, it took me a minute, but I made it work.  (Oh, the things you do for your children!)

Then we got to work.  We drew and we drew and she whistled a happy tune (literally) for over twenty minutes.  She would not let me look at her work.  She was in heaven.  This is what she finally unveiled to me:

To me, it's glorious.  Even though it's not a factor tree, per se, it shows so much of how her 7 year old self is thinking about numbers, including an emerging understanding of factors.  All the numbers she wrote are obscured by the coloring, so I had to ask her about them.  At its base  are the numbers 10, 10 and 20.  Notice the overall symmetry of the tree.  The flowers at the base are 5+5+5+5 (do you see the 'plus' sign written on the trunk?) and, even better, the colors are inverted in each pair.  Moving up the tree, the next two longer branches on the outside are 2 and 10 (so 10x2) and the two small branches inside them are both 10 (so 10+10) and the crowning flower is 20!

This tree motif is doing so much more for her visualization of numbers than those little cards ever could!!  I'm so excited.

And, as a little bridge to future trees, after she walked me through her drawing I showed her mine.  I asked her what she noticed and what the differences were between the trees. 

She noticed a lot, but not the numbers so I pointed her attention to the fact that although each tree split in different ways (10 and 2, 4 and 5), they still had the same prime flowers -- two 2s and a 3. (Did you notice I made each prime it's own color?  And I wonder if there are other ways to bring out the number properties beside just writing the numeral?)

There is no need for us to rush into full-out factoring, but it is an interesting exploration and variation on the division we've been doing.  And, the introduction of another type of number, the primes, is an interesting development as well.  I think this is an approach that we will be able to use time and time again: create, observe, discuss, posit, repeat. 

Let's see: trees as metaphor.  I've seen fractal trees, factor trees, and trees whose branches are the positive numbers and roots the negative ones.  There is so much potential in this kind of image/metaphor for number explorations through all elementary grades.  And, you do know that part of this approach is giving kids an empty piece of paper, some awesome pens and letting them create their own trees from the ground up, right?

I know one thing for sure: our number wall is going to be gorgeous!!  [Addendum: Here's what happened the next day.]

All-in-One? Using Shapes to Explore Number & Algebra Concepts

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. 

Survival Math -or- How Cuisenaire Rods Contribute to Independent Living



From South to North: Our House, Bloomington, Park,
Nashville, Columbus (IN), Ohio (see the archway?)

Darling girl is an early riser and a morning learner.  Every day she rises, recites her list of questions and plans, and sets it all in motion.  This morning the dear child drew herself a map.

Soon after, she was really, really frightfully intent on writing number sentences.  All before my first cup of caffeine.  Normally I'd just let her chatter away and let it all wash over me (it was only 7am after all) but today the tide was just too strong.  I didn't really understand why there was such a need to do all this before breakfast but sometimes you just need to act first and ask questions later. 

She had already started her 'math lesson' by writing down 1+1=2.  Then 2+2, 3+3, 5+5.  At some point she said, "Every time I add, the numbers get bigger!"  The answer to 5+5 was doubled, which was doubled again.  80+80 gave her some pause, but by then I had drunk my tea and decided we might as well get out those Cuisenaire rods again. 

I wasn't completely sure if she was ready to move on to labeling each rod with a unit number or, for that matter, if I really understood how to use the rods to represent addition, or if we were going in the 'right' sequence of learning with these things.  But why let any of that stop me?

I said, "First, can you just put the rods in order from smallest to biggest?"  Done.  "Okay, let's look at your first number sentence, one plus one.  Here are two of the white blocks.  Which block is the same length as two white blocks?"

She picks up the red rod.  I say, "Okay, how many whites makes a red?  So when you write 'two plus two' this is what it looks like.  Which rod equals four whites or two reds?"  She pulls the purple rod to match.

 

 Then I had a bright idea:  "Can you figure out how many white blocks make an orange?  What other ways can you combine the rods to make ten?"  Girl interrupts my line of thought, "I want to make a line of a hundred!" She knows that ten tens make one hundred so she starts stacking them together to make sure she has ten of them before she lines them up.  I persist, "How many more ways can you make a ten?"

 

She wanted to see how long a line the ten tens would make when lined up end to end. 

This is how long it was and there are apparently many different ways to combine the rods to make ten.  We didn't get to all of them, I'm sure.  Then she wanted to see what a line 200 units long would be...



"Look Mama!  I'm 110 units long!"

Overall, my biggest takeaway from this morning is that, although she has enough questions to keep us busy for now, I need to do some more research on the best way to use Cuisenaire rods -- quick!

Later in the day I did find out why she was so intent on doing math this morning.  It's actually a secret so if you see her don't let on that you know.  "Mama," she said, "You know why I've been wanting a longer math lesson and have been so interested in maps and money?  Because T. [a good friend] and I were wondering what would happen if you [the mommies] stopped loving us and those things would be important to know."

Apparently, she and her great friend T. have hatched a sort of secret survival plan for an undefined point in the future when "our mommies don't love us any more."  They've decided they need to know how to do math, figure out how to work with money, and know how to survive in the outdoors.  After I assured her repeatedly that I will ALWAYS love her, we went to the library to get some books about how to identify and cook with edible plants. 

She said, "Don't tell T. that I told you.  And even if you never stop loving me, it's still helpful information to know."

"I won't tell," I replied, "But it's probably good that I do know so I can help you find the information you need. Learning all this will be really helpful when you grow up and live on your own."

"But I'm going to live with you forever."

Exploring the Nature of Numbers

I've written elsewhere in this blog about how I am remediating myself by learning math along with my currently five year old daughter.  I am a classic case of someone who probably would have LOVED math but instead got confused by a worksheet in kindergarten (something about bumblebees) and never fully recovered.  In high school I loved geometry so much that I was motivated to memorize all the theroms.  I was never able to apply what I learned, but I think it was the challenge and process of understanding interesting rules that caught my attention.  Plus, geometry is visual, which is my thing too. 

I really had to address my math phobia when I came up with the bright idea to integrate math with percussive dance, but what I realize now is that my love of geometry (which some people, I've heard, call the only 'real' math, but that's another argument conversation) has really served me in that effort.  And, the effort I've put in over many years to make sure I really understand the math the fits with percussive dance has also restored some of that natural mathematical curiousity which I lost back in kindergarten in the '70's.  I am really starting to see how math can be playful and full of interesting ideas and questions and it's own form of inquiry unto itself, just like being playful in the process of creating percussive patterns.  It's okay to ask questions that have more than one answer.  Not only that, it's FUN!

So, anyhow, this morning my daughter was teaching her toy cat Matilda her letters and some basic arithmetic.  They do other kinds of math in her kindergarten, but she seems taken with what numbers are and how they get bigger and smaller.  Here is a snapshot of our conversation this morning.  First she tells me:

"I'll write 100 really BIG because it's a BIG number and 1000 even BIGGER. And then a quadrillion."

This is really interesting to me, actually.  I do, despite what I've said so far, understand the concept of scale and size and amounts, etc.  But, I read somewhere that the whole 100 days of school project (which practically every kindergartener experiences) started back in the 1980's and I'm sorry I missed out!  On the 100th day of school this year she brought in 100 dried chickpeas in a little baggie.  I was surprised that it actually didn't look like very much.  She also, as a joke (and this was totally her idea) brought:

"One hundred pennies in the form of a dollar bill." 

My daughter has also had some questions about if the numbers ever end.  I think she was happy when we read a book called The Cat in Numberland which was about the Hotel Infinity where numbers keep moving in and there are always enough rooms for them, and it seemed to provide the answer she was looking for.  She still tells me she loves me 'to the end of the numbers' which is very sweet, even if it's not accurate!

In her kindergarten class she's been learning basic addition and subtraction.  She comes home and writes down little equations.  This morning she said, "Tell Matilda [her toy cat] what subtraction is."  I said, "What can you tell her?"  She answered, "When you take away a number to make a smaller one." 

Also on the subject of subtraction, at some point recently she started trying to take away a larger number from a smaller one, which led me to comment, "Did you know there are numbers smaller than zero?"  I tried to explain but I didn't know if I made sense or if she understood, until this morning.  She was writing down an equation for her cat.  This is what she said while she wrote it out:

"Three minus eight equals....zero!  And I'll put a minus sign on the zero because it's less than zero."

My response?

"I think I'll go out and get you a number line!"