What does = mean?

There's a lot of work out there on how american kids don't know what = means.  Nicole McNeil, Martha Alibali, and Bethany Rittle-Johnson have done some really remarkable work on this phenomenon recently.

It boils down to all of the arithmetic problems that kids see in an american math class have the form:

• 3 + 4 = ____
• 4 + 29 = ____
• 54*31 = ____
• 81/ 9 = ____
• 1/4 * 2/3 = ____
  

Are you starting to see a pattern?  There's a computation to be performed then an equals sign, then a blank to be filled in.  If this is the only type of math problem that you see then clearly = means 'do the calculation'

From this point of view, the statement 4+5 = 3+6 makes no sense.  There's no computation to be performed.  So it's just confusing.  If you believe that equals means 'write the answer' and there's no answer to write, then that's going to create some cognitive dissonance.  And when students see a problem like 4+5 = 2+ ___, they can't even understand the problem. 

The thing is, this misconception about what = means doesn't go away by high school, or college.  There are even some studies that show that practice like the items in the bulleted list actually make students even worse at mathematics, because they reinforce that misconception about equality.  Students don't understand that equals means 'the two sides are the same' because all of their experience says that equals means 'put the answer'

This is a well studied problem, but it's closely related to something I observed yesterday in a student that came to my office hours.  The class is a 200-level stats class for science and engineering majors.  This guy came in for help studying for today's exam.  He kept asking, so What do I do with a Binomial distribution? what do I do with a Poisson distribution?    I'm not sure I can explain the level to which these questions don't make sense.  It's kinda like asking, What do I do with an elephant?  You could feed the elephant, you could ride the elephant, you could train the elephant to do tricks, you could weight the elephant.  There's no computation that you do on an elephant, it just is.  You can use the elephant to accomplish different tasks, but there's no one thing that an elephant is for.

The circus has come to town

So, the Prez and some 19 of his buddies are having a party across the street from my office tomorrow night.  I've been walking past armies of suits on my way to school every morning. But don't worry, the lapel pins are always in place to let you know what country the suit came from.  Apparently my office building is closed because it's in sniper range or something.

This is the brand new 8 ft pop-up fence that's been installed since yesterday.

distributed leadership

There's a view of leadership that it's heroic.  A great leader takes the reins and drives the herd somewhere new and exciting against all odds, fighting forces from within and without, and the exceptional things that are achieved rest solely upon the leader.

Now this might work for a general, but a principal who has this view is supremely annoying.  I really like the work of Jim Spillane out of Northwestern University on distributed leadership.  In this view, leadership is setting up the organization so that everyone in it is working toward the same goal effectively.  The actions of the leader tell less than half the story, the key is really the interactions.

Which is why I am so annoyed at Mowgli's school principal again.  They have space to set up a sensory room in the K-2 autism support classroom.  They have the space!!!  This is the number one thing that most schools don't have.  They need a little bit of money to get the space set up right.  I volunteered to help raise the money.  Apparently, they're so un-used to volunteers that this has made their heads explode.  They don't know if we can donate money or if we have to donate the objects.  They don't know if I could bring a team of volunteers in to help paint the room, they have to check with the maintenance union.  The fact that I want to make a donation to the school, and they're reaction is 'wow, I don't know if that's allowed' rather than 'wow, let's do this thing' blows my mind.  

Of course this is on top of the fact that I suspect Mowgli's fork got confiscated at the metal detector yesterday.

They Might be Giants strike again

All sorts of awesome!

Teaching Play

One of the things we've been working really hard on around here is pretend play.  This is a really hard thing to teach because most children don't have to be taught, they just do it naturally.  One way to try to teach it is to give the child some templates of play, or play scripts.  Since Mowgli can already read, I decided to use books to provide the templates.

I went to the bookstore to try and find a few appropriate books, but the selection, um.... left something to be desired.  All of the books at his reading level had plots that were way way way to complicated.  All of the books for younger kids had no plot at all.  So I decided to make my own.

They were really fun to make.  They've both got a total of 4 pages including the cover.  They're bound with embroidery thread.  The illustrations were done with india ink & watercolor.  I constructed 2 more books that I haven't written anything in yet.  I'll post them as soon as I've finished the illustrations. 

Let's Play Dragon

 

Let's Play Pirates. 

Mowgli loved them & they've been great so far.

Images are free to copy for use in therapy. 

An altered state of normal

I thought about just jumping in.  Maybe showing off my latest creation.  Writing a post like what I hope will become average fare around here.  But then I figured an introduction is more in order.

See I don't really know what normal looks like any more.  Every family has stories about the time cousin dave jumped into the swimming pool when he was 2 and Aunt Sarah had to go in after him.  My family has stories about the time Mowgli almost walked off the bow of the USS Stewart at Sea Wolf Park.  

Notice the destroyer is parked on concrete, not docked in water.

But see those events don't really sum up the true distance from normal.  For most people, those are outlier events.  Things that happen once or twice in a childhood, they're not things that you have to plan against constantly.  We have alarms on all of our windows because more than once, Mowgli has gone out of a 3rd story window.  And I don't mean he's leaned out.  I mean all the way out, standing on a ledge about 5 inches wide.  Why not lock them?  Well the boy's just too smart, he's figured out every lock we've put on every door or window.  So we settle for alarms.

So what does normal look like around this house?  Normal is asking, "What do you want for dinner?" and getting "coming soon to DVD" in reply.  Normal is driving Mowgli to some sort of therapy 3 days a week, and having a therapist visit our house on 3 of the remaining 4 days.  Normal was crossing the street to avoid someone walking their dog so that Mowgli didn't become terrified of the pup and go into fight-or-flight mode.  Normal is trying to teach a little boy who just started first grade and is reading at a second grade level how to play with a toy pirate ship.

Think about that.  You don't have to teach kids how to play with toys.  You put the toy in front of the kid and imagination takes over.  But that's life in our house.  We have to teach Mowgli how to play.  We have to teach him how to say 'catch' before he throws a ball and pegs another kid.  We've worked very hard to teach him to say "I don't like that."  The boy will eat tofu, and while shuddering from how badly it tastes, say, "I like tofu." 

This dragon is called autism.  We just do our best to keep dancing. 

 

Mowgli