Saturday, March 31, 2012

Fun with Functions: To Infinity and...Back!

What is a function?  Before yesterday I had no earthly idea except that it's clearly an important mathematical concept and one that young children can easily experience at the concept level.  Based on my search for function machine activities last night, it seems that functions are essentially about rules and change, both of which seemed to dominate my little experiment with the kid today.

It all started yesterday when I was looking for math games at the library and found the book Anno's Math Games II.  The first chapter is about a marvelous, magical machine that turns one thing into something else.  A chicken into a chick, a butterfly into a caterpillar, and...can you guess the rule?   Later in the chapter Anno moves the rules toward transforming numbers but when I suggested to my six year old that we make a magical machine of our own she jumped up and literally shouted:

"Yay!  Let's make one!  I'll go get Lucy [our cat] and we'll turn her back into a kitten!"

Um.  Well, not really a real machine, right?  Sort of a machine in our heads?  With numbers?  We left it there for the day.

Last night I got to searching.  It seems that a function machine is actually pretty straightforward.  You put a bunch of numbers in and try and figure out from a number of results what 'the rule' (function) is.  So, I made my own function machine using clip art of a washing machine and a dryer (dirty in and clean out...) and some arrows.  I figured that we could play around with it a little and see what happens. 

What happened was that it basically flopped.  I made the first rule, we put a number in and wrote down what came out.  We did it a couple times and then she successfully guessed what the rule was.  Not exciting.  Then she got to make up a rule, and the whole thing collapsed into an activity that essentially boiled down to practicing familiar math facts.  Plus, she was really, really disappointed about the machine itself.  I think partially because it was a picture, and we all know a picture of a machine is not a real machine, right?

No worries, though.  At that point she took matters literally into her own hands and made her own...

There were buttons for adding, subtracting, multiplying and dividing.  There were buttons for stop, go and compute. 

There was a little door where you could put in a slip of paper with the input written on it.  There was an emergency brake. (If you can't tell by now, these were all her ideas.  I was just grateful she had salvaged the activity -- even if it meant being demoted to the position of scribe as she dictated the rules of how to operate the thing.)

There was also a screen: "Numbers can be a bit dangerous," she said. "The screen is for if you don't want to go on into infinity." There was a steering wheel "in case we go into infinity, or a really, really hard problem.  Press the steering wheel to pause, so you can think about it.  Or, if you get into infinity steer the wheel backward and you'll be back where you started."

Apparently, the transformation of numbers is not only tricky but a bit dangerous as well.  

When it was time to try out the machine I encouraged her to put her results on my nice little worksheets, but her energy was really more about the slips of paper she wrote on and put into her machine.  Fair enough.  Here are some of the rules we played around with:

One round of multiplying by three.  Four rounds of 'multiply by 2, take away 1' (on another sheet, not pictured).

We also put the machine on 'geometry setting' and steered "the wheel backward to get it to change the number of sides" of a shape.  We started with a triangle and the rule of 'add one side'.  Interestingly, the kid turned the triangle into a rhombus, not the square I expected, which made a lot of sense given the angles.  And, when she tried to add a side to the square she initially just divided the square with a diagonal line.  A nice moment to reinforce 'side' and 'edge'. 

Also, because we recently explored the concept of half-ness and double-ness we did some doubling using the legos.  I suppose we were on the 'geometry' setting then, as well, because of the 3D shapes.

Reading this post over, I realize I forgot one really important idea: that you can run the 'rule' both forward and backward through the function machine.  Except that...the machine she built does have a steering wheel.  As she did say, " can steer the wheel backward and you'll be back where you started."  I guess that's an inverse operation right there!

I've been thinking about how we might do this next time.  Harnessing my kid's imagination and her need for the concrete/physical seems like the key.  Maybe we'll figure out how to make bigger machines, some toy-sized, some human-sized.  The little plastic cats could go in one door and a different amount of bigger stuffed cats could come out the other?  Or, a dress-up function where you go into the machine (a tent?) dressed normally and come out with hats, scarves or necklaces added?  Hmmm...  Maybe we could make up simple Jump Patterns (ala Math in Your Feet) and change them somehow? 

It's easy enough for me to come up with ideas for physical/concrete functions, but harder to think of ones that create some complexity and/or interesting results. I also can't tell which is more important -- doing the computation of the rule, or figuring out what the rule is.  I've obviously got some thinking and learning to do here so if you are able to elucidate this topic for me, I'd be really grateful!

In the end, it's clear from watching her make her own machine that she's basically got the concept: you can change something into something else using different kinds of rules.  More importantly (for her, anyhow) is that you don't have to settle for a representation of a machine -- you can make your own real one!

Wednesday, March 28, 2012


You are nine (or ten, or eleven). You have power.

You think mathematically while you move.  You make your own choices about how far to turn and in which direction, what kind of movement to use, where to put your feet, how to combine your patterns and whether to transform your pattern with symmetry.

Your choices are good, especially when they take time to perfect. 

You can enjoy the process...

...or not.

You can have fun and work hard at the same time.

You can be the one who usually gets sent around the building 'for laps' because you can't focus in class, and yet, in this case, you are the one who makes up the most fantastical, awesome pattern ever.

You marvel at how fast an hour goes.  Where does that time go?

It goes into your feet.

You have agency over your body, your brain, your ideas, and your learning.

Now that is power.

[It's also images of Math in Your Feet in action!  Photo credit: Ryan Richardson.]

Tuesday, March 20, 2012

A Game of Her Own: Discovering Division

This was one of those moments where you ignore the phone, turn off the stove, and just pay attention.

I was making a grid for a sight word bingo game, like the one above except still blank. My kid looked the blank chart over and said,

"I'm going to make a game board like that, and I'm going make it a math game."

Curious, I replied, "Okay. What's your game?"

"You divide an even number in half and then add it to another number."

It didn't actually turn out the way she had described, but she did make her own math game!  How cool is that!?  I should say that we both knew going into this that we were 'in development' with the game, that we really wouldn't know how to play it until we were done. 

The game is a bit convoluted but patterns did emerge.  She dictated the directions to me which I've put below.  I'll fill in the details later in the post.

really flat, stiff paper
permanent marker
drawing stuff of all kinds like crayons or something
that's about it

Step One: Make the Grid
Have the mama make the grid, 25 squares, five rows, five columns.

Step Two: How to Play the Game
Write all even numbers in the grid starting with two.

[After filling it about half way she exclaimed, "Hey, this is interesting!" and pointed out that each column included the same numeral in the one's place.]

Step Three:
Pick any number on the grid.  Use the pennies to divide that number in half.  Here's an example: One half of 28 is 14.  Color them both the same color on the grid because we know they're related.

The Kid's Final Words:
By playing this game we'll learn how to multiplicate (sic!) and we'll also learn our two times tables.

The Mama Adds:
By playing this game you'll also get a sense of 'even-ness' and 'double-ness' and 'half-ness' not to mention support your emergent abilities in division.

So, as you might imagine, this was a verrry interesting process to watch.  I should say that it was probably inspired by a game of tic tac 10 and tic tac 15 we played on Sunday.  She used the even numbers for those games and it probably reminded her that an even number can be divided into two equal parts.  That's the interesting thing -- she knew that in concept but not really in practice, until she made up this game.

The first number she chose was 18.  What's half of 18?  She couldn't do it mentally, so I suggested some Cuisenaire rods which she immediately deemed 'too confusing'.  I pulled out the change jar and she got to counting out the pennies, which turned out to be a perfect manipulative.

Counting out eighteen pennies seemed a little confusing at first, for some reason.  I casually mentioned that it helps me to put things in groups and then skip count, and that it's easy to count by fives.  One slim pause and she took the bait. 

What really fascinated me was watching her trying to figure out how to divide the pennies into two equal groups.  First she spread out the pile of eighteen pennies into something that approximated two separate groups.  From there she echoed the strategy I had suggested, above, and put the pennies into rows, some four pennies long, some five.  I can't remember how the rest of it played out, but at some point the rows were all the same amount and there were two extra pennies without a clear home.  For some reason she didn't (couldn't?) just split them up, one to each group.

I think what she was looking for was some 'even-ness' in the rows as well as the total amount in each group.  She moved the pennies around until, voila!  She had figured it out visually but we also counted just to make sure that we had even amounts in each pile.

I suggested writing the answer in the corner of the 18 box, just so she'd remember, and she continued to make notes in this manner for the rest of the game.  We looked but couldn't find '9' anywhere else on the chart.  Why?  Because there are no odd numbers on the even number chart, silly!  Time to pick another number! 

This time she chose 28.  Since we still had eighteen pennies out, I asked how many more pennies she needed to make 28.  With the answer she added the extra pennies into the mix and proceeded to divide them into two groups more easily than before.  I'd show you a picture of the process but she told me to stop taking pictures!  It was almost dinner time and she was focusing so hard, she said the camera distracted her.

Anyhow, half of 28 is fourteen, which we did find on the chart.  I suggested coloring the 14 in and she suggested coloring both numbers and using the same color so we'd know which numbers were 'related'. 

And then we were both worn out, it was time to make dinner and so we left it there for the day.  I wasn't sure if she'd want to keep going the next day, but she did.  On day two of making/playing this game there was a really interesting progression in her ability to figure out 'half' of most numbers on the chart.

The first number she chose on day two, 36, required the use of pennies to figure out and, again, she wanted the pennies to line up evenly in their rows.  While placing all 36 pennies in rows of five she said, "I think there's going to be an extra one..."  She played around with lining them up so that all the rows were the same length and, luckily, these are even numbers, right?  So it did eventually work out that way, as you can see in the picture above. 

For the next number I suggested 20 and encouraged her to figure out the half in her head.  Easy!  It was at this point that things really took off.  She was able to figure out half of 48, 24, 12 and 6 all in her head by her own strategy of dividing the tens and then the ones 'in half', and then adding the resulting ten(s) and ones together.  Pretty cool!

Then there was the 32, 16, 8, 4, 2 progression.  She had just figured out half of 16 and I wondered what would happen if she went the other way and doubled 16.  She went immediately to pencil and paper (seen above) for this which is interesting since she usually sticks to mental math and/or her fingers.  She didn't complete the sums with 'regrouping' (something she's not learned yet) and instead used the written numerals to help her thinking, something that sounded like "Okay, there are two tens and...[thinking]...thirty two!"  

All told, she used four strategies in this game to help her figure out half and/or double of the even numbers: manipulating pennies, dividing in her head by addressing tens and then ones, adding (sort of) on paper, and accessing basic addition and multiplication math facts she's internalized already.

The colors for the squares in the chart are for numbers that are 'related' as she termed it.  2, 4, 8, 16 and 32 are all pink because they are a sequence of doubling.  6, 12, 24 and 48 are all red for the same reason.  We did a final check of the squares that had not yet been colored in.  Turns out halving each created an odd number which, obviously, were not on the chart.

She was so excited and proud about making her own game.  As for me, I'm marvelling at how beautiful it is when a child literally constructs her very own understanding of math.  Neither of us really knew where the process of creating this game was going to take us, but it was obviously a worthwhile journey.  Thanks for sharing it with us!

Wednesday, March 14, 2012

Multidisciplinary Mathematician: Sonya Kovalesky

I ran across this quote and link (below) on a forum I frequent.  I found it fascinating to read about a mid-19th century woman with a genius for mathematics and the kinds of road blocks she faced to practice her 'art'. 

And, because I will probably not become a mathematician myself but am still endlessly curious about what it means to do and think math, I am greatly appreciative of anyone inside mathematics who is able to express it's basic nature (and beauty) so clearly for the rest of us.

"Many who have never had occasion to learn what mathematics is confuse it with arithmetic, and consider it a dry and arid science. In reality, however, it is the science which demands the utmost imagination [which is more than just making things up] ..... It seems to me that the poet must see what others do not see, must look deeper than others look. And the mathematician must do the same thing. As for myself, all my life I have been unable to decide for which I had the greater inclination, mathematics or literature."

For me, this is just one more piece of evidence that there are more connections and similarities between disciplines than there are differences.  This is especially true when the focus is on creating meaning rather than building a set of technical skills.  And, in my mind, this connectedness is a good thing.  Kids who get to learn under this assumption show great enthusiasm and effort because it's real.  They know the real world is connected, not broken up into indiscriminate, arbitrary pieces.  When subjects are taught as parts of 'the whole' and not as isolated chips of knowledge, everything makes more sense.  Math taught out of context or without beauty or with a single minded focus on procedure, well...doesn't. 

You can read a short and fascinating biography of Sonya Kovalesky, 1850-1891, here.

Sunday, March 11, 2012

Is it Cheating to Use the Multiplication Chart?

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...

Wednesday, March 7, 2012

Channeling Tana Hoban: Juxtaposition Edition

When my kid was a preschooler, I used to stare wistfully at the Tana Hoban books at the library.  Because my kid rarely liked any stories outside of a narrative context, I knew there was no way she'd sit down with me to take a closer look.  Turns out, the thing I love more than looking at Tana Hoban's work is pretending to be Tana Hoban.

It sort of just happened, but the best things are like that, I think.  The plan was to do 'cheetah training' at the Y and then explore campus on an extraordinary morning full of sun and blue.   We completed the first part of the plan and started the second.  We really had no agenda but then things started catching our eyes.  The first one was the building with the intricately carved limestone decorations and the round tower four stories high.  What would it be like inside?  Should we go explore?  Yes!

Here's something we saw on the outside of the building:

Turns out that if it's round on the outside, it's round on the inside, too.  How do you lay square tiles in a curve?

And the view from the round tower was just as round.

After a full exploration of Maxwell Hall (including an incredible ten foot limestone fireplace and a very mysterious and steep spiral staircase) we decided to continue our explorations outside.  The funny thing was that, even though we've spent a lot of time on this part of campus, we kept seeing things we'd never seen before...

Like hexagons!  Gorgeous ones!

Teeny tiny squares in a grid, inside a circle!

Hexagons in circle, around a pentagon!

A basket weave of parallel lines, in a circle!

A drain with wavy, curvy lines!

My personal favorite, a tessellation of rhombuses!  You don't see too many rhombuses out in the wild like this. 

Then it was time to got back to our car and go home for lunch.  On the way, we found even more designs, especially ones that expanded from a center point into a circle.

Flowers, too.

Concentric circles.  "And stars!"

A curve through the squares. 

And more.

So much wonderful design in this modest drain.

Trash can.

"Look Mama! More circles!"  Notice, more curves.  So lovely.

My favorite thing was finding shapes within shapes and especially so many different kinds of patterns with both straight and curved lines. This is a great example of what Maria Drujkova of Natural Math calls 'growing your math eyes.'  Once you've got 'em you see math everywhere!

If we go 'out looking' again, I'm going to pack two cameras, and some paper and crayons for making rubbings. 

Monday, March 5, 2012

All in Good Time

The kid and her dad are in the other room.  I can hear them -- he's teaching her how to play the penny whistle.  She's had tin whistles in her music box for years, playing around with them as she will, but today, apparently, she asked him to give her a lesson.  It's also the first time that she's ever asked anyone to 'show' her how to do something.  She's six years 'and over a half' as she tells it -- is this what Piaget meant by 'concrete operations'? 

Her dad started with one note, which requires one finger.  The second note, two fingers.  Three fingers down.  One to two to three.  "I can go backwards," she tells him.  They've been working on covering the holes completely and breath control for forty minutes.

"You're a good teacher!" she told him. 

There are a couple things going on here that fascinate me. 

The sounds of the scale, or even part of it, are ordered.  There's a clear forward and backward to it, especially while fingering the tin whistle, and you have audible feedback when you do it 'in order' or play around with combinations of notes. 

I love when kids think about doing something in reverse.  It happened last Friday on the last day of Math in Your Feet up at a Greenwood school.  It happened this weekend in my house.  Any time you recognize or create a pattern, whether it be visual, kinesthetic, musical or number-based, you are thinking mathematically.  Any time you are changing a pattern, especially on your own volition, you are thinking mathematically.

I'm also fascinated by her sudden switch from playing around with the whistle to specifically asking for 'a lesson' and being able to focus on that lesson for over an hour.  It's a great example, to me anyhow, that kids learn when their brains are ready.  And, from what I've observed as both a parent and a teacher, every brain seems to be on its own timetable.  It's easy for me as the mom to be anxious about certain skills not showing up (reading especially) when I think they should, but really? 

Really, I think I need to be more patient.  And, in the future, when I've got some silly idea that she 'should be able to x, y or z by now' I should just pause and recall all the other times it's cycled like this for her -- seemingly 'fallow' periods followed by an explosion of new skills and interests, like what happened this weekend...

The kid in question played her whistle non-stop for the last 24 hours and is now one note shy of a full scale, can play those notes forward and backward, and has learned the classic 'Hot Cross Buns'.   Oh, and she added up multi-digit UNO score this morning with the six cards she won from me by adding them into 15 and 14 then adding the result up in her head and was able to tell me how she solved it.  She's memorized half of the song "Fifty Nifty United States" and decided to write down as many state names as she could remember.  Plus, in the last 45 minutes before bedtime last night she wrote her own cat-themed creation story on the typewriter and just brought me ten multiplication equations written down in pencil, although the numbers are still being written backwards and in reverse order.  "How'd you do that?" I asked.  "I counted them up on my fingers, all by myself!  Isn't that cool?" 

It's a veritable brain 'storm' around here.  I'm not sure why I worry.


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