## Saturday, May 5, 2012

### Sidewalk Math: Functions!

After a mild winter we had a lovely and quick blooming spring which allowed us to get out and about earlier than we might have otherwise.  In March I posted about an outside adventure where we discovered a veritable treasure trove of circles in juxtaposition with other shapes.  I think this might have been the origins of what I've started calling 'sidewalk math'.

Sidewalk math is fun because, generally, all you have to do is keep your eyes open.  If you've got a camera to record your observations, all the better.  This is not necessarily an original idea; the photographer Tana Hoban has a whole series of books with photos of the math all around us.  Her camera is the eye through which we can notice math in the physical world.  There are also the engaging Math Treks developed by Maria Droujkova of Natural Math.

For us, sidewalk math is a combination of these two approaches and has turned into a large percentage of our first grade math classroom.  It capitalizes on my daughter's propensity to notice everything, fulfills her need for movement while she learns, and bypasses her resistance to formal lessons.  It's also an opportunity for us to make observations and pose questions in a collaborative way, which is an approach that works for both of us.  For example, on a recent walk my daughter notice a crack in the sidewalk that initiated an hour-long in-depth conversation and exploration into the nature of triangles as we traveled to the hardware store and back.

(And it's apparently it's sticking with her: As I'm writing this my daughter calls down to me to report that she and her dad saw "seventeen triangles on their way home from the park this morning....did you know that part of an arrow is a triangle?!" )

But, in this story, sidewalk math plays another role, that of salvaging my initial attempt to introduce functions to my young daughter.  You can read about my first attempt here where she was wholly and unequivocally unimpressed with my presentation of the subject and took matters into her own hands.  I ended the post wondering what to do next.

I was understandably thrilled when I came across the book A Game of Functions by Robert Froman.  It's part of the Young Math Series from the 1970's and is out of print.  A quick Google search found copies available for purchase between \$17.00 and \$115.00!!  Luckily, my husband works at a university with a very comprehensive library and I got my hands on a copy.  I read it to my daughter one morning.  She wasn't having a great day, but she didn't protest, and we got through most of it.  I let the idea sit, waiting patiently for an opportunity to put the ideas into action as the book suggests.

The book starts out with an introduction to the idea of 'function', as in 'whether we go the park this afternoon is a function of the weather -- if it rains this afternoon we will go shopping, if it is nice we will go to the park' (I'm paraphrasing here).  Or, as in this example below, how long it takes you to run around the outside of your house depends on on whether you crawl, walk or run. How quickly you go is a function of your mode of movement.

At that point, the book introduces the functions 'game'.

You find a nice big area and draw a line across and a line up.  Lucky for us I had sidewalk chalk on me and we were at a park with a parking lot that looked almost like graph paper!

When I asked the kid what her 'rule' was, she said she wanted to take ten steps over and ten steps up.  We quickly realized that we needed a way to make sure her steps were the same length so we landed on her personal foot length, heel to toe.  She made a little white chalk X at ten steps.

And then, ten steps up from the X, and marked with chalk.

Although I helpfully informed her that she didn't need to go back to the beginning each time to 'add ten' to the last result, it was interesting watch her ignore me completely and then figure it out for herself.  And it didn't take her long -- by the time she was working on 30 steps, she realized she should just add ten to the second X (twenty steps) instead of count 30 from zero.  It wasn't an 'I told you so' kind of moment, just a little bit more proof that if the kid wants to figure something out on her own I should just let her do it. Lesson learned and internalized!  (For her and me.)

When she had used as much space as she could I asked her to stand in the corner and look at all the Xs she had marked up into the space.  "They go in a diagonal!" she observed.  And then she ran from (50,50) all the way to (0,0).

We also worked on another rule for a little while (nine out and eight up) but she was running out of steam.  That was a lot of thinking for one morning.  It was perfect timing, too.  As we were packing up, four cars drove into the parking lot and covered her work!

At the very least, I feel like I've redeemed this concept for her (or, more likely, myself).  I haven't labeled what we did 'functions' but I did use the word 'rule' a lot, for example "The rule is to 'add ten' so your next move is ten more steps than the last time...let's see what happens when you do that a bunch of times!"

As you can see, above, the book goes on to show how you can do the same work on graph paper.  I'm thinking about how to make it a game...maybe two rolls of the dice determine the rule?  I could do my line and she could do hers and then we could compare?  Steepest line?  Line with the most graphed points?   Which one gets to the top of the paper with the least number of graphed points?  And, maybe include the question: "How steep your line is depends on (is a function of)....?"

Who knows?  This journey is full of adventure and surprises.  It's not always smooth sailing, but we're learning a lot, her and I.  And, one thing's for certain, there is more sidewalk math in our future.

1. Thanks for sharing this example of learning the function concept.

Hung-Hsi Wu says that from grades 5 to 7 the learning of abstractions in math has to pick up to prepare for algebra. Your daughter appears to be considerably younger than this. This support Wu's view that grade 5 is already time to teach more abstract ideas than just numbers.

This is in his talk, "From Arithmetic to Algebra".

2. I love this game! I might try it with my son although we'll have to scale down and reduce the step counts. One of the ideas that comes to mind (and that worked for us) was to tape out a grid on the floor using blue painters tape. Then have a toy car (ours was a lunar rover) drive to a predetermined point (i.e. 3 up and 3 across) to pick up rock samples and leave a flag.

3. Yelena, it's definitely a flexible game. I love the idea with the painters' tape (I love anything that involves painters' tape, actually ;-). Maybe we'll try that next -- I love the narrative that accompanies the activity that you did with your son. Not sure what ours would be. Hmmm....

This version/form of the functions concept seems to work better for my daughter than the machine/numbers one, for some reason. It's more visual I think and it tapped into her strong kinesthetic needs which allowed us to do the game without a narrative structure.

We did take the game to graph paper and dice a couple days later. I have a new possible post in my head about that.