The Map is Not the Territory

For some reason, I thought that when we stopped homeschooling and my eight year old started going to 'regular school' (albeit an independent community school) I would have less access to her mathematical thinking and learning.

Turns out I was wrong.  Two examples:

Walking to school.  The girl is holding her library card like it's made of gold.  She thinks maybe they're going to the downtown library that afternoon and she doesn't want to part with it.  We pass a brick wall built with some of the bricks on the diagonal, creating an interesting pattern.

She stops to notice.  "Look at this cool design!"

She notices something else. "Hey, the library card is the same width as the brick."

And, another thing. "This brick is two and a half....no, two and a fourth times longer than the card."

[All thanks, I think, to the great afternoon we spent last winter with the book Ten Times Better - read about our antics here.]

Second example, math homework, and she actually wants my company.  If you know our history, the fact that she invited me into her math world is truly a momentous occasion.

The homework is titled "Things That Come in Groups".  There are eight juice boxes in a package. She easily figures out how many juice boxes in three packages.

Next question asks how many juice boxes are in six packages.  And....wait for it...she says, "Well, there are 24 in three packages so there are 48 in six packages," and proceeds to write that equation in the spot it's requested.  The next question is about nine packages, and she continues the pattern of reasoning -- 72 total juice boxes.

I suppose I get delighted by little things but, no matter, this is wonderful to watch.  I know what kind of equation they were asking for, though.  It actually took her longer to figure out how to to write it the way they wanted (6x8=48) but no matter.  I love that she saw patterns and easily computed the answers in her head. And, because I know her math learning history, I remember her at age six developing this strategy as she made her very own game about doubles and halves.



"I love homework, mama."

"Why is that?"

"It feels so old fashioned and classic."

I look forward to more math homework because I think it will afford many more opportunities for watching her mathematical thinking expand and deepen.  I love watching kids learn, especially so up-close. Sigh.

Every day this week we've been playing with math dice. Enthusiastically.

I'm not going to name the company because not only do I not review or endorse any product on this blog for money or power (not that they asked) but it is also quite easy to go out to your local games shop and get your own set of two 12-sided and three 6-sided dice. (The rules are also pretty easy to figure out: multiply or add the two numbers on the 12-sided dice and then roll the six-sided dice and try to find a way to make the target number using as many operations as you know.)

Did I mention the enthusiasm?


My newly eight-year-old is enthusiastic about many things but has always been a little standoffish with her affinity for math, probably because, I think, she perceives it as my 'thing'. So, it's been nice to be able to truly enjoy a math game together.  (It's been a while -- we were heavy into UNO a couple years back which was super fun.)  It's clear my kid is on her way to a happy relationship with operations, but there's something even more interesting developing...

I had always thought my girl was not what I would call 'systematic' or 'precise'.  I know for sure she is prone to intuitive leaps of connection or understanding and lots of messy tinkering, none of it looking either precise or systematic to my eyes.

As I've been drafting and revising this post I've realized that maybe she has been those things, I just haven't been able to see it.  And, as we've been playing the dice game I've watched her systematically running through different combinations of the six-sided dice (by moving the dice physically to different positions) and  reasoning to herself out loud as she thinks through the different ways to use the hand she's rolled.

I guess I always thought that precision in mathematical problem solving looked, well, neat and orderly and on paper.

Anyhow, I am not (too) ashamed to admit that I was wrong. I think she's been precise and systematic in her own way for a while now. In retrospect, I realize I've heard this kind of  'talking herself through' a series of moves or ideas before. Systematically. In math and in many other contexts. For years. In a messy, verbal, highly enthusiastic way.

Okay, so I'm a slow learner I guess, but pretty open minded all the same. I think it's worth considering that there must be a difference in the way children and adults go about their reasoning. Or, at the very least, that I have a deeply ingrained image of 'what it looks like to do math'. I'm going to keep thinking about all this. If you have any observations or resources to share on this subject, I'd be tickled pink.

In the end, I'm super impressed that not only is she beating the pants off me but she has also created her own strategy for combining operations to reach a target number.  And it's all her.  The only thing I did was bring out the dice.

So...who likes bedtime?  I, for one, am not generally enamored with detours from the normal bedtime routine.

But when my kid said, "Hey Mama! Want to hear this song I made up that helps you with your three times table?!" I decided to give her a little leeway so I could capture the moment.   What can I say, I'm a sucker for math!

The song starts: "There were three ice cream trucks at the corner of Circle Drive..."



My favorite part is 'on the corner of Circle Drive' since, of course, circles have no corners!  But, I'm pretty sure this was not intentional on her part.

Did see her looking off to her right to silently skip count the answers in her head?  We've been doing a lot with conceptualization of multiplication/division (arrays, multiplication towers, factor dominoes, scale, re-imagined factor trees, exploring the concept of units) but almost nothing with memorization.  

I'm happy to have this unexpected piece of evidence that she is thinking about and internalizing these concepts.  I'm also pleased to report a happy conclusion to bedtime!

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.

We were at our second classroom (aka local co-op cafe) and just starting our project when a guy comes over and enthusiastically asked, "Can you tell me what you're doing?"

"Oh," said I, "we're building a multiplication tower."

"A multiplication tower?  What's that?"

"Well, we've got these posts on a grid, and we are going to use different colored beads in different amounts to..."

And then he says, I kid you not, "Oh, yes, multiplication is just repeated addition, really."  (Seriously, I'm not making this up!)

I was flummoxed, because after some great conversation about multiplication in the comments section of a post of mine from last March, and the aha! moment I had about scale while reading the Ten Times Better book, I know that's not all there is to multiplication.  Not at all.  I was so flummoxed it was all I could do to reply, "Well, it's also about understanding scale and measurement and rate..."

And just like that he cuts me off and says, "Well, this seems to be a good concrete way to learn addition..." (or something like that) and leaves, obviously much less enthused and impressed than when he came up to us.

And that about sums up the collective view of elementary mathematics, doesn't it?  Essentially, the message we get is: "There's nothing much to it, just learn your facts and when you get a little older you can do real math [pat on the head]." 

No, I'm not bitter, just perplexed, because when we started building this tower the questions started flooding in.  I started wondering about a LOT of things, most of all the commutative property (which I inadvertently keep calling a process, and I think I might be on to something, see what you think.)  Here is what we did:

I measured out a grid and my daughter and I inserted bamboo skewers into the intersections to make a total of 25 posts.  Our first try at the tower grid was to start with one bead at the origin and then move outward on both the x- and the y-axes with two beads on the next posts, three beads on the next, etc.  Here's a picture (isn't it pretty!?): 



First, it's such a nice three-dimensional gradient, don't you think?  And the different colors show the multiples of each amount clearly.   But the questions started when we got through with 2 x 2 = 2 green + 2 red.  Moving up to (1, 2) I suddenly thought:

"What colors should we use?  Is that two 3s or three 2s???"  It helped to see that the square numbers 4, 9, 16 and 25 moved up the diagonal and it calmed me a little to know that, whatever happened, we weren't completely on the wrong track.

It also helped to turn the model and look at it from different perspectives.  This is a nice view because you get a linear progression of the colors.  On the far left the posts increase by five: 1 five, 2 fives, 3 fives... and on the far right the posts increase by one: 1 one, 2 ones, 3 ones... This, incidentally, is the view my 7 year old prefers, perhaps because it is so orderly.




































But then I started wondering, what if every column from (1, 0) onward stayed the same amount all the way up to (1,4) and so on for (2,0), (3,0) etc.?  If you look closely at the picture below you will see that that means instead of two 3s there are actually three 2s and instead of two 5s, there are actually five 2s.  It changes the look....







































It's a similar result to attempt #2 but the progression to purple now occurs on the x axis!  My daughter likes this version a lot, but I'm not so sure.  The whole thing is less symmetrical than the others but, on the other hand, there are other things to observe and learn from it that you don't see in the first two.




I'm sure there is more analysis I could do, including comparing the total number of beads and distribution of color used in each  attempt but that's for another day.  My point is, yes, multiplication and addition can be done in any order, but our response should not be 'so what, that's easy, don't forget, let's move on' but rather, "Wow, that is SO amazing!!!"

I mean, sure, you get the same final answer no matter what, but just look at the variation in quality.  For example, a combination of four 2s (2 green, 2 red, 2 blue, 2 yellow) looks completely different from two 4s (4 green, 4 red).

Before I started my personal math remediation, multiplication seemed like nothing more than a table of right answers.  Now, however, it seems like a wonderful opportunity to find as many different right answers to the same question as possible.  My kid got a little bored with the beading process but no matter.  I'm going to leave my favorite version out on display, which will make at least four different multiplication models up in our house right now, for us to ignore or puzzle over as we will.  It's already generated some nice conversation about square numbers. 
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p.s. I got inspired to do this project while reading the proofs of the grids chapter in the new Moebius Noodles book.  They say the project's target age is birth to age 6 but, using today's activity as an example, I really think anyone can benefit from their playful, hands-on, inquiry based approach to math. 

p.p.s. I kept the skewers long during the first beading iteration but quickly decided to trim them because I truly could not see the forest for the trees, lol!  I think the trim helps focus the eye on the gradient and the patterns produced by the color quantity and progression..

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!

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. 

I know it doesn't look like much, but there was a battle going on.  We rolled the dice (one six sided, the other twelve sided) and staked out our territory.  We're warrior cats, you know.  You didn't?  Have you read those cat clan books?  My kid hasn't -- too many kitties getting hurt makes her crazy -- but her best friend has.  They 'play' those stories every time they meet.  You should see them in their 'warrior training'.  It's crazy.

Since success in this game was mostly left to chance, I knew I had to think quickly. "Let's stake out our clan territories and see who gets control of the most area!" I exclaimed.  Needless to say that's all the motivation she needed.  She decided almost immediately that each new piece of territory had some kind of real-life correspondence -- campus (the scene of our recent Tana Hoban adventure), our favorite hangouts, she wanted them all.  I pointed out that she could be strategic with where she filled in her boxes and, if you look at the first photo again, you'll see that she's surrounded parts of my blue territory so that I have to cross her land to get to other parts of mine.  Nice.
With her watching, I took the yard stick and marked off a 1/2" grid over the entire city.  The grid was not to scale but was consistent across the entire map.  The area unit was 'one square'.
We looked at the map and found our landmarks -- hangouts, parks, campus, lakes, our neighborhood -- and at the map's key to find the roads, schools, bike and walking trails, etc.  Then we started strategizing about which areas we'd most like to 'have control over'.

Source: Math is Fun

I found the kid (still six) writing down little equations, I guess just for something to do.  She had written "9 x 8 = ____" and was staring at the blank space.  

In the last couple weeks she has developed her own strategy for finding answers to equations like these.  She does it by adding it all up on her fingers.  We did the roll-the-dice grouping game a bunch of times over the course of a month (January, I think?) and she seemed to catch on to what it was all about.  But now, it seems, counting on her fingers works just fine except that when she gets to numbers over six or seven it can take a long, long time to find the answer.  She's not noticeably perturbed by the effort, and from my point of view it's awesome that she understands that she's counting nine, eight times to get the answer. That's the main point of what multiplication is at this stage, right? .

Although, there she was, still trying to figure out the answer. So, I offered, "How about I go get the multiplication table?" She agreed but when I brought it back she said, "But I don't know how to use it!"

Visually tracking columns and rows is difficult, even for an adult sometimes, so I hit upon two strategies, which I showed her:

The first is to put your pencil in the first column, in this case the 0, and leave the tip just above the nine row, then move the pencil column by column until it's in the right one for the equation, in this case the eight. The tip points directly at your answer.

The other strategy is to just find the nine in the shaded area on the left and, using your finger, point to each box in that row and say "zero, one, two, three..." until you get to eight and there's your answer. Easy, right? But the girl had a different reaction:

"THAT'S CHEATING! If I do it this way, how am I going to really understand it????!!!"

I must admit I was a tiny bit impressed with this statement but I was also more than a bit flummoxed about how to respond. I mean, what's there to really understand? You learn what multiplication means (which I think she's got), you learn your facts and then you use them when you need them. I finally said:

"Well, addition is the most important thing to know how to do in your head. When you add in your head you are learning how numbers combine and recombine to make other numbers. It's an important skill to have and that's why they don't have a facts chart for it. It's not quite the same for multiplication -- what people usually do with the times table is memorize it. There are all sorts of fun number patterns to find in this chart..."

At this point she was still absolutely convinced that using the chart to find any answer constituted some kind of unlawful activity. Trying a different approach, I said, "Okay, I have an idea. If you really want to understand how and why multiplication works, you can skip count. You already know how to skip count zero, ones, twos, fives and tens. So all you need to do is learn how to do that with threes, fours, sixes, sevens, eights and nines. It's like adding..."

That was the best I could do in the heat of the moment. My daughter is prone to ginormous reactions and sometimes it's hard for me to think clearly under duress.  But, the question still remains -- is it 'cheating' to use the multiplication chart?  I should add that she seems to have no interest in memorizing the facts either. 

After writing this all out I'm starting to think that that maybe all this (her finger counting strategy, eschewing the chart, etc.) is just her way of mastering the content.   All the same, I am wondering what kind of understanding she wants.  Maybe she's looking for familiarity?  Maybe at some point she'll finally figure out on her own that memorizing the times tables facts is actually pretty helpful?  Am I missing anything here?

Sigh.  If you have any thoughts, including helpfully pointing out any errors you find in my explanations and reasoning, I'd love to hear them...