Thursday, October 13, 2016

Making Connections

One of the things I really appreciate about the CCSS New California State Standards is that we want students to make connections across domains and grade levels. And, while some may disagree with me here, I appreciate the transferrablilty of the standards for mathematical practice. Things like attending to precision, constructing viable arguments, critiquing the reasoning of others, looking for and using structure, and problem solving in general all play in other content areas life.

Any chance I get to make a connection to another area, I do it.  I read an article a while ago by Hung Hsi Wu where he treated a variable as a pronoun.  It made sense to me.  It makes sense to my students, so we go with it.

We are about to dig into equations and expressions, but I really can't stand how textbooks approach this.  You get to see maybe one or two simple expressions that may be tied to a context, but then a million exercises with expressions so complicated, there's no way a kid can tie it to anything that matters.

So here's where pattern problems come in.  Fawn has done a tremendous service for us.  I'm also really digging Dudamath lately because I can be more intentional with the patterns I put in front of my students. Seriously, if you haven't played around with this site, go there now.  It's pretty amazing.

We've done a few pattern problems and we are getting the hang of doing the generalization, but writing an expression has been more difficult.  So, here's where Wu helped.

Our morning announcements just mentioned our volleyball team won yesterday, so that provided a nice context.

Maria played volleyball.
Shanay played volleyball. 
Teresa played volleyball. 
Jan played volleyball. 
Jill played volleyball.

I wrote these sentences on the board and asked if they could write one sentence that captured the essence of all the others.

"Maria, Shanay, Teresa, Jan and Jill played volleyball."


"Maria and her friends played volleyball."

which eventually became

"She played volleyball."

Right next to the sentences, I wrote the following expressions:

3 + 1
3 + 2
3 + 3
3 + 4
3 + 5

and it didn't take long for us to settle on some version of 3 + n.

The groups then went to work on today's pattern problem.  The use of some sort of variable when trying to describe a rule made it's way into their work.  Many of the groups are still in progress, but movement was made today.  Let's see how it goes tomorrow.

Monday, October 3, 2016

Building Fraction Sense

The struggle here is real.
A student has no idea where to place a fraction on a number line (because fractions aren't numbers, of course) but can convert to a mixed number like a champ.

My attempt to help out:

This applet gets at the heart of the things I've enjoyed working on lately.  The initial estimate offers very little help, but as the student progresses through, they have more references which allow the revisions to become more precise.  When my students worked with this applet, there were audible groans when I asked them to lower the lids on their computers as well as exclamations of "I got it!" when they moved closer to 0% error.

Here's a GeoGebra book that goes from estimating fraction to addition to multiplication.  I'm still working on division, but that should drop soon.

Tuesday, September 13, 2016

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.

Wednesday, September 7, 2016

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?  

Friday, September 2, 2016

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.  

Monday, August 29, 2016

Math Don't Break

Integer operations are always an interesting endeavor with 7th grade students because they come pre-loaded with so many rules.  So. Many. Rules.

We've been talking about making our own rules, so we have this sequence of products and I ask students to discuss what patterns they notice.

-3 (3) = -9
-3 (2) =  -6
-3 (1) = -3
-3 (0) =  0
-3 (-1) = ??

Stuff we noticed:

"It starts with a -3 every time."
"It goes down by 1."
"It changes by 3."

I zero in to the apparent contradiction in going down by 1 and changing by 3 so we can clean up the language a bit.  This starts an nice little exchange about whether or not going from -9 to -6 is an increase or decrease.  We conclude it's actually an increase.  I have to remember to take my time here because this isn't an insignificant point:  Kids seem to think in absolute value.  

So what comes next? 

I wrote down everything I heard.  

"3".  "-3".  "4".  "-4".  

"Wow!"  I say.  "We've got a great argument about to happen.  This is awesome!  So many different opinions.  So which is it?"

Some minds change when groups start to discuss.  The students who thought 4 or -4 were thinking of sums and not products.  That leaves 3 or -3.  

"Ok, so which is it?"

If I had a dollar for every time a student said "A negative times a negative is a positive" followed by "because my teacher told me", I'd have all the dollars.  

But then Isaac offers a reason worth looking at. 

"I think it's -3, because positive 3 times positive 1 is positive 3, so negative 3 times negative 1 is negative 3."

So I write the following on the board:

(pos) (pos) = pos
(neg) (neg) = neg

We talk about this pattern Isaac. has noticed.  "Does this work for you all?"

Jordan speaks up, "I don't think so.  It has to be positive three so that it doesn't break the pattern."

"Which pattern is that?"

"The pattern goes from -9 to -6 to -3 to 0.  It's increasing by 3 each time so the next answer has to be 3."

"Why would that be so?" I ask. 

Then Vanessa chimes in.

 "Because math don't break."  

Thursday, August 25, 2016

Strategy vs. Procedure

I really want to focus on students being mindful of their process.  What they are doing is important, but they really need to know why they're doing it.  We've been doing daily exercises, How Many Squares?  that are based on Michael Fenton's activity, How Many Peaches?

We usually highlight different student strategies and have spent some time developing a continuum of strategies that looks something like:

counting --> grouping/adding --> skip counting --> multiplying --> writing/evaluating math expressions

This student's particular strategy generated a nice conversation.

I asked whether or not students thought this was a strong strategy.  Responses were less than enthusiastic so it was time to move a little.

Me:  Alright, if you think this is a strong strategy stand on this side of the room;  if you think it's not move to the other.

It was 31-2 in favor of the strong.  So I walk over to the "not strong" side and make my case.

Me: It can't be a strong strategy because the answer is 84 and this student said it was 76.

About half the class moves to my side.  I figure it was an even split on who was convinced by the "right answer" argument and who was convinced by the "I'm your teacher" argument.

Two students on the strong side raise their hands.

Student 1:  I think it's still a strong strategy because he probably just made a mistake.

Me:  Probably?  Where does that fall on our argument continuum, gut level, some reason or convincing reason?

Student 1:  Some reason.

Me:  Ok, great.  Can anyone take it to the next level?

Student 2:  I think it's still a strong strategy because he just counted 11 instead of 12 across the top.  He still multiplied right, but he just used the wrong numbers.  Everything else was good.

Yeah, that'll play.