When Your Good Friends Don't (But Should) Get Along

I swear I'd give CalcDave's left arm (you're right handed, right Dave?) in order to be able to embed a GeoGebra applet into a Desmos activity.  I mentioned it, once or seven times, but that's right about the time the Desmos customer service director seems to drive into a tunnel.

I mean, I don't hate this activity or anything.

But I hate this slide.

Stupid Math Notation

Sometimes students show a misconception that makes me pause and wonder how we can continue without clearing this up.

Sometimes the misconception isn't their fault.

Take the "-" symbol for instance.  Are we talking about subtraction?  Negative numbers?  How about "the opposite"?  Or inverse; maybe it's inverse.

I gave students this number line today with the prompts.

1.  Tell me everything you can about the number P.
2.  Show where -P is on the number line. Tell me everything you know for sure about -P.

They did very will with the first prompt.  Lots of responses like:

"P is on the negative side."
"P is a negative number.  It's between -2 and -3."
"P is probably about -2.7 because it's closer to -3 than it is to -2."

Ok, I'm loving this.  Then they drop the hammer on me.

"-P is negative."
"-P is also on the negative side."
"-P has a negative sign in front of it so it's also negative."

What are your first steps when you encounter thing like this?  

The (Selfish) Reason I Keep Teaching

Pick a teacher's blog.  Go ahead, pick one.  Go through the archives and you'll likely find a post talking about vocation or calling or some other noble reason to enter the profession.  You'll also find some variation of the phrase "I don't teach subjects; I teach children." These are all true, but I don't think they get at why I teach.  I mean, I'm no Mr. Shoop and, while, I do think there's  satisfaction to be found in helping others, I'm not quite ready to side with the Tribbianian philosophy of good deeds. What I am willing to admit is that one of the things that keeps me teaching is a little selfish.

Let me explain.  When I was in high school, I took one of those aptitude tests.  The results of that test told me I should either be a teacher, a  youth pastor or, yes you guessed it, a cab driver.  At first, I was thinking, "Cab driver?  What's that about?"  But as I thought about it, these three career paths have something in common:  people.  So, then why teaching?  I'm going to try to impact people no matter what I do.  So why teach?

That brings me to the selfish reason:  Teaching is a case study in why people do what they do.  I'm really interested in that.

Dan recently asked about the motivation for moving away from the text book when lesson planning. I think this gets at why I'd rather do my own thing even though I didn't realize it when I first responded.  I want to know why kids do what they do, and most textbooks can only expose what they do.  If I make my own activity, I can ask the questions the way only I ask them. It's my way of starting the conversation with my students.

Today was a great example of this.  We were working through a Desmos activity where kids had to model sums using a number line.  (I really wanted them to be able to sketch on an interactive graph, but, whatever, can't have everything.)  But this activity exposed two really important misconceptions. One I was very aware of and the other I had never considered.

Misconception #1

I've seen this one before.  Often times students count the numbers and not the spaces.  Ok, got it.  I know how to deal with this. 

Misconception #2

At first glance I thought I had this one pegged too.  Students are just stating the length of the segment.  Through the discussion, however,  it came out that a significant number of students said the blue segment represented positive three because it was on the positive side of the number line.  

In 20 years, I've never seen that.  

It led to a nice chat about direction and location and how these can influence the value of a number.  

I don't have this conversation locked down.  And that's why I want to come to work on Tuesday. 

That and I want to see if Desmos has those sketchable interactive graphs yet.